Cote n° 78 · pages 6–128
· 382 displayed formulas · [Polygones réguliers et polynômes cyclotomiques] : notes manuscrites (s.d.).
Inventory dating : [à partir de 1977]
Édition de démonstration
\[(1) \qquad E = \mathbb{C}\]
LaTeX source
\[
(1) \qquad E = \mathbb{C}
\]\[\text{\struck{\ill{}}}\,(2) \qquad \zeta^n = 1, \quad \zeta^d \neq 1
\ \text{si}\ d \mid n,\ \text{\struck{\ill{}}}\, 1 \leq d < n\]
LaTeX source
\[
\text{\struck{\ill{}}}\,(2) \qquad \zeta^n = 1, \quad \zeta^d \neq 1
\ \text{si}\ d \mid n,\ \text{\struck{\ill{}}}\, 1 \leq d < n
\]\[(3) \qquad s_i = \zeta^i \qquad i \in \mathbb{Z}/n\mathbb{Z}\]
LaTeX source
\[
(3) \qquad s_i = \zeta^i \qquad i \in \mathbb{Z}/n\mathbb{Z}
\]\[(4) \qquad
\begin{cases}
\sigma z = \bar{z} \\
u z = \zeta z
\end{cases}
\qquad \text{\struck{\ill{}}}\]
LaTeX source
\[
(4) \qquad
\begin{cases}
\sigma z = \bar{z} \\
u z = \zeta z
\end{cases}
\qquad \text{\struck{\ill{}}}
\]\[(5) \qquad
\begin{cases}
\sigma s_i = s_{-i} \\
u s_i = s_{i+1}
\end{cases}\]
LaTeX source
\[
(5) \qquad
\begin{cases}
\sigma s_i = s_{-i} \\
u s_i = s_{i+1}
\end{cases}
\]\[\bigl(\mathcal{T}/\{\pm 1\} \smallsetminus \{\bar{1}\}\bigr)\]
LaTeX source
\[
\bigl(\mathcal{T}/\{\pm 1\} \smallsetminus \{\bar{1}\}\bigr)
\]\[(s_{i+3} - s_i) = \text{\struck{\ill{}}}\, r\,(s_{i+2} - s_{i+1})\]
LaTeX source
\[
(s_{i+3} - s_i) = \text{\struck{\ill{}}}\, r\,(s_{i+2} - s_{i+1})
\]\[r = 1 + \underbrace{(\zeta + \zeta^{-1})}_{2 \cos \arg \zeta}\]
LaTeX source
\[
r = 1 + \underbrace{(\zeta + \zeta^{-1})}_{2 \cos \arg \zeta}
\]\[\frac{\zeta^{i+3} - \zeta^{i}}{\zeta^{i+2} - \zeta^{i+1}}
= \frac{\zeta^3 - 1}{\zeta^2 - \zeta}
= \frac{\zeta^2 + \zeta + 1}{\zeta}
= 1 + (\zeta + \zeta^{-1})\]
LaTeX source
\[
\frac{\zeta^{i+3} - \zeta^{i}}{\zeta^{i+2} - \zeta^{i+1}}
= \frac{\zeta^3 - 1}{\zeta^2 - \zeta}
= \frac{\zeta^2 + \zeta + 1}{\zeta}
= 1 + (\zeta + \zeta^{-1})
\]\[u\,(\operatorname{Tr} v\,\mathrm{id} - v) = (\operatorname{Tr} v)\,u - uv\]
LaTeX source
\[
u\,(\operatorname{Tr} v\,\mathrm{id} - v) = (\operatorname{Tr} v)\,u - uv
\]\[\operatorname{Tr} u\check{v} = \operatorname{Tr} u \operatorname{Tr} v
- \operatorname{Tr}(uv) \qquad 2 \operatorname{Tr} \ill{}\]
LaTeX source
\[
\operatorname{Tr} u\check{v} = \operatorname{Tr} u \operatorname{Tr} v
- \operatorname{Tr}(uv) \qquad 2 \operatorname{Tr} \ill{}
\]\[\det(u + v) = \det u + \det v
+ \underbrace{(\operatorname{Tr} u \operatorname{Tr} v
- \operatorname{Tr} uv)}_{\varphi(u, v)}\]
LaTeX source
\[
\det(u + v) = \det u + \det v
+ \underbrace{(\operatorname{Tr} u \operatorname{Tr} v
- \operatorname{Tr} uv)}_{\varphi(u, v)}
\]\[\begin{pmatrix} \lambda & u \\ 0 & \lambda' \end{pmatrix}
\qquad
\begin{cases} u x = 0 \\ \operatorname{Tr} u = 0 \end{cases}\]
LaTeX source
\[
\begin{pmatrix} \lambda & u \\ 0 & \lambda' \end{pmatrix}
\qquad
\begin{cases} u x = 0 \\ \operatorname{Tr} u = 0 \end{cases}
\]\[\begin{aligned}
\operatorname{Isom}(\Pi, \Pi') &\longrightarrow \operatorname{Rep}(\Pi') \\
\theta &\longmapsto \theta(r_0)
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\operatorname{Isom}(\Pi, \Pi') &\longrightarrow \operatorname{Rep}(\Pi') \\
\theta &\longmapsto \theta(r_0)
\end{aligned}
\]\[\theta \mapsto \theta(r) \colon \operatorname{Isom}(P, P') \to
\operatorname{Rep}(E', S', \omega')\]
LaTeX source
\[
\theta \mapsto \theta(r) \colon \operatorname{Isom}(P, P') \to
\operatorname{Rep}(E', S', \omega')
\]\[\text{\struck{$\Gamma \xrightarrow{\varphi} \operatorname{Aut}_{\mathrm{aff}}(E)$}}\]
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\[
\text{\struck{$\Gamma \xrightarrow{\varphi} \operatorname{Aut}_{\mathrm{aff}}(E)$}}
\]\[\Gamma \subset \operatorname{Aut}_{\mathrm{aff}}(E)\]
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\[
\Gamma \subset \operatorname{Aut}_{\mathrm{aff}}(E)
\]\[(1) \qquad \Gamma \simeq D_n \qquad
\text{(isomorphisme non donné)}\]
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\[
(1) \qquad \Gamma \simeq D_n \qquad
\text{(isomorphisme non donné)}
\]\[(2) \qquad S \in \text{\struck{\ill{}}}\,\Gamma \backslash E\]
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\[
(2) \qquad S \in \text{\struck{\ill{}}}\,\Gamma \backslash E
\]\[(4) \qquad s_0 \in E\]
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\[ (4) \qquad s_0 \in E \]
\[(4) \qquad \omega \subset \Gamma\]
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\[ (4) \qquad \omega \subset \Gamma \]
\[(5) \qquad
\begin{cases}
\operatorname{card}(\omega) = 2, \quad \omega = \{u, u^{-1}\} \\
\text{\struck{\ill{}}}
\end{cases}\]
LaTeX source
\[
(5) \qquad
\begin{cases}
\operatorname{card}(\omega) = 2, \quad \omega = \{u, u^{-1}\} \\
\text{\struck{\ill{}}}
\end{cases}
\]\[(6) \qquad
\begin{cases}
\text{\struck{\ill{}}}\,\operatorname{card} \Gamma/\Gamma^{0} = 2 \\
\Gamma/\Gamma^{0} \ \text{opère sur}\ \Gamma^{0}\ \text{par}\
\lambda \mapsto \lambda^{-1} \\
\text{l'ext.\ } \Gamma\ \text{de}\ \Gamma/\Gamma^{0}\ \text{par}\
\Gamma^{0}\ \text{est semi-directe,} \\
\quad \text{i.e.}\ \exists\, \sigma \in \Gamma,\ \sigma^2 = 1,\
\sigma \mapsto \text{él.}\ {-1}\ \text{de}\ \Gamma/\Gamma_0
\end{cases}\]
LaTeX source
\[
(6) \qquad
\begin{cases}
\text{\struck{\ill{}}}\,\operatorname{card} \Gamma/\Gamma^{0} = 2 \\
\Gamma/\Gamma^{0} \ \text{opère sur}\ \Gamma^{0}\ \text{par}\
\lambda \mapsto \lambda^{-1} \\
\text{l'ext.\ } \Gamma\ \text{de}\ \Gamma/\Gamma^{0}\ \text{par}\
\Gamma^{0}\ \text{est semi-directe,} \\
\quad \text{i.e.}\ \exists\, \sigma \in \Gamma,\ \sigma^2 = 1,\
\sigma \mapsto \text{él.}\ {-1}\ \text{de}\ \Gamma/\Gamma_0
\end{cases}
\]\[(9) \qquad \text{\struck{\ill{}}}\
\begin{cases}
S\ \text{n'est pas contenu dans une} \\
\text{droite de}\ E
\end{cases}\]
LaTeX source
\[
(9) \qquad \text{\struck{\ill{}}}\
\begin{cases}
S\ \text{n'est pas contenu dans une} \\
\text{droite de}\ E
\end{cases}
\]\[(8) \qquad S\ \text{est un torseur sous}\ \Gamma^{0}\]
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\[
(8) \qquad S\ \text{est un torseur sous}\ \Gamma^{0}
\]\[(7) \qquad \Gamma^{0}\ \text{fini}\]
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\[
(7) \qquad \Gamma^{0}\ \text{fini}
\]\[\Gamma \hookrightarrow \Gamma', \qquad \Gamma^{0} \simeq \Gamma'^{0}\]
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\[
\Gamma \hookrightarrow \Gamma', \qquad \Gamma^{0} \simeq \Gamma'^{0}
\]\[(7') \qquad \operatorname{card} \Gamma^{0} \simeq \mathbb{Z}/n\mathbb{Z}.\]
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\[
(7') \qquad \operatorname{card} \Gamma^{0} \simeq \mathbb{Z}/n\mathbb{Z}.
\]\[\text{c}') \qquad S \in \Gamma^{0} \backslash E\]
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\[
\text{c}') \qquad S \in \Gamma^{0} \backslash E
\]\[\text{\struck{(8$'$)}} \quad \text{\struck{$\Gamma^{0}$ opère fid.\ sur $S$}}
\quad \text{\struck{$u \in \omega$ fixés, si $s_0 \in S$, $u s_0 \neq s_0$}}\]
LaTeX source
\[
\text{\struck{(8$'$)}} \quad \text{\struck{$\Gamma^{0}$ opère fid.\ sur $S$}}
\quad \text{\struck{$u \in \omega$ fixés, si $s_0 \in S$, $u s_0 \neq s_0$}}
\]\[\text{\struck{(8$''$)}} \quad \text{\struck{$\operatorname{card} S = \operatorname{card}(\Gamma^{0}) = n$}}\]
LaTeX source
\[
\text{\struck{(8$''$)}} \quad \text{\struck{$\operatorname{card} S = \operatorname{card}(\Gamma^{0}) = n$}}
\]\[(10) \qquad \exists\, \sigma_{s_0} \in \Gamma,\ \sigma_{s_0} \neq 1,
\ \text{tel que}\ \sigma_{s_0} s_0 = s_0\]
LaTeX source
\[
(10) \qquad \exists\, \sigma_{s_0} \in \Gamma,\ \sigma_{s_0} \neq 1,
\ \text{tel que}\ \sigma_{s_0} s_0 = s_0
\]\[(11) \qquad
\begin{cases}
\text{a}')\ \text{Représentation \struck{fidèle}}\quad
D_n \xrightarrow{\ \varphi\ } \operatorname{Aff}(E) \\
\text{b}')\ s_0 \in E
\end{cases}\]
LaTeX source
\[
(11) \qquad
\begin{cases}
\text{a}')\ \text{Représentation \struck{fidèle}}\quad
D_n \xrightarrow{\ \varphi\ } \operatorname{Aff}(E) \\
\text{b}')\ s_0 \in E
\end{cases}
\]\[\begin{aligned}
D_n &= \{\pm 1\} \cdot \mathbb{Z}/n\mathbb{Z} \qquad
(\sigma \in \{\pm 1\},\ u \in \mathbb{Z}/n\mathbb{Z}) \\
&= \{\sigma, u \mid \sigma^2 = 1,\ u^n = 1,\
\sigma u \sigma^{-1} = u^{-1}\} \\
&= \{\sigma, \sigma' \mid \sigma^2 = \sigma'^2 = 1\}
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
D_n &= \{\pm 1\} \cdot \mathbb{Z}/n\mathbb{Z} \qquad
(\sigma \in \{\pm 1\},\ u \in \mathbb{Z}/n\mathbb{Z}) \\
&= \{\sigma, u \mid \sigma^2 = 1,\ u^n = 1,\
\sigma u \sigma^{-1} = u^{-1}\} \\
&= \{\sigma, \sigma' \mid \sigma^2 = \sigma'^2 = 1\}
\end{aligned}
\]\[(13) \qquad
\begin{cases}
\text{\struck{$u s_0 \neq s_0$}} \\
(1^{o})\ \sigma s_0 = s_0 \\
(2^{o})\ s_0,\ s_1 = u s_0,\ s_2 = u^2 s_0\ (= u s_1)\
\text{non alignés}\ [\Rightarrow u s_0 \neq s_0] \\
\text{\struck{\ill{}}}\,(12)\ \varphi\ \text{fidèle}\
(\text{P.\ mémoire})\quad u\ \text{unimodulaire}]
\end{cases}\]
LaTeX source
\[
(13) \qquad
\begin{cases}
\text{\struck{$u s_0 \neq s_0$}} \\
(1^{o})\ \sigma s_0 = s_0 \\
(2^{o})\ s_0,\ s_1 = u s_0,\ s_2 = u^2 s_0\ (= u s_1)\
\text{non alignés}\ [\Rightarrow u s_0 \neq s_0] \\
\text{\struck{\ill{}}}\,(12)\ \varphi\ \text{fidèle}\
(\text{P.\ mémoire})\quad u\ \text{unimodulaire}]
\end{cases}
\]\[s_0 \notin D_\sigma \cap D_{\sigma'}\]
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\[
s_0 \notin D_\sigma \cap D_{\sigma'}
\]\[k^* \simeq \mathfrak{G}\]
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\[
k^* \simeq \mathfrak{G}
\]\[D_\sigma^{*} = D_\sigma \ \text{est un}\ \mathfrak{G}\text{-torseur}\]
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\[
D_\sigma^{*} = D_\sigma \ \text{est un}\ \mathfrak{G}\text{-torseur}
\]\[\text{\struck{\ill{}}}\ \mathfrak{G} \subset \mathfrak{G}\
\text{\struck{\ill{}}}.\]
LaTeX source
\[
\text{\struck{\ill{}}}\ \mathfrak{G} \subset \mathfrak{G}\
\text{\struck{\ill{}}}.
\]\[\begin{aligned}
\varphi &\colon D_n \longrightarrow \operatorname{Aff}(E) \\
\varphi' &\colon D_n \longrightarrow \operatorname{Aff}(E')
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\varphi &\colon D_n \longrightarrow \operatorname{Aff}(E) \\
\varphi' &\colon D_n \longrightarrow \operatorname{Aff}(E')
\end{aligned}
\]\[v_E = \mathrm{id} \iff v(s_0) = s_0 .\]
LaTeX source
\[
v_E = \mathrm{id} \iff v(s_0) = s_0 .
\]\[u^{i} s_0 = s_0 \implies i \equiv 0 \ (n) \qquad (\text{i.e. } u^{i} = 1),\]
LaTeX source
\[
u^{i} s_0 = s_0 \implies i \equiv 0 \ (n) \qquad (\text{i.e. } u^{i} = 1),
\]\[v_E = \mathrm{id} \iff v(s_\nu) = s_\nu \quad \forall \nu\]
LaTeX source
\[
v_E = \mathrm{id} \iff v(s_\nu) = s_\nu \quad \forall \nu
\]\[n = n' d\]
LaTeX source
\[ n = n' d \]
\[\varphi' : D_{n'} \longrightarrow \mathrm{Aff}(E)\]
LaTeX source
\[
\varphi' : D_{n'} \longrightarrow \mathrm{Aff}(E)
\]\[\begin{cases}
\varphi' : D_{n'} \longrightarrow \mathrm{Aff}(E) \\
s_0 \in E
\end{cases}\]
LaTeX source
\[
\begin{cases}
\varphi' : D_{n'} \longrightarrow \mathrm{Aff}(E) \\
s_0 \in E
\end{cases}
\]\[(14)\qquad
\begin{cases}
\sigma_E,\ \sigma'_E \in \mathrm{Aff}(E) \quad \text{satisfaisant} \\
\sigma_E^{2} = \sigma_E'^{2} = \mathrm{id}_E ,
\quad (\underbrace{\sigma'_E \sigma_E}_{u_E})^{n} = \mathrm{id}_E
\end{cases}\]
LaTeX source
\[
(14)\qquad
\begin{cases}
\sigma_E,\ \sigma'_E \in \mathrm{Aff}(E) \quad \text{satisfaisant} \\
\sigma_E^{2} = \sigma_E'^{2} = \mathrm{id}_E ,
\quad (\underbrace{\sigma'_E \sigma_E}_{u_E})^{n} = \mathrm{id}_E
\end{cases}
\]\[\begin{cases}
\sigma_E s_0 = s_0 \\
s_0,\ s_1 = u_E s_0 \ \text{et}\ s_2 = u_E^{2} s_0 \ \text{sont non alignés}
\end{cases}\]
LaTeX source
\[
\begin{cases}
\sigma_E s_0 = s_0 \\
s_0,\ s_1 = u_E s_0 \ \text{et}\ s_2 = u_E^{2} s_0 \ \text{sont non alignés}
\end{cases}
\]\[(15)\qquad \forall \text{ diviseur } d \mid n, \ \text{on a} \ u_E^{d}(s_0) \neq s_0\]
LaTeX source
\[
(15)\qquad \forall \text{ diviseur } d \mid n, \ \text{on a} \ u_E^{d}(s_0) \neq s_0
\]\[(\sigma'_E \sigma_E)^{n} = \mathrm{id}_E\]
LaTeX source
\[
(\sigma'_E \sigma_E)^{n} = \mathrm{id}_E
\]\[(16)\qquad \varphi : D_\infty \longrightarrow \mathrm{Aff}(E)\]
LaTeX source
\[
(16)\qquad \varphi : D_\infty \longrightarrow \mathrm{Aff}(E)
\]\[(17)\qquad \forall v \in \mathrm{Aut}(\Pi), \quad \exists !\, v_E \in \mathrm{Aut}(E)\]
LaTeX source
\[
(17)\qquad \forall v \in \mathrm{Aut}(\Pi), \quad \exists !\, v_E \in \mathrm{Aut}(E)
\]\[(18)\qquad
\begin{cases}
\text{a)}\ \ \varphi_\Gamma : \Gamma = \mathrm{Aut}(\Pi) \longrightarrow \mathrm{Aff}(E),
\quad v \longmapsto v_E \quad \text{homom.\ de groupes} \\
\text{b)}\ \ S \xrightarrow{\ \varphi_S\ } E
\end{cases}\]
LaTeX source
\[
(18)\qquad
\begin{cases}
\text{a)}\ \ \varphi_\Gamma : \Gamma = \mathrm{Aut}(\Pi) \longrightarrow \mathrm{Aff}(E),
\quad v \longmapsto v_E \quad \text{homom.\ de groupes} \\
\text{b)}\ \ S \xrightarrow{\ \varphi_S\ } E
\end{cases}
\]\[(19)\qquad S_E \ \text{n'est pas contenu dans une droite}\]
LaTeX source
\[
(19)\qquad S_E \ \text{n'est pas contenu dans une droite}
\]\[(20)\qquad \sigma_0 \neq 1_\Gamma , \quad \sigma_0 i_0 = i_0\]
LaTeX source
\[ (20)\qquad \sigma_0 \neq 1_\Gamma , \quad \sigma_0 i_0 = i_0 \]
\[(21)\qquad
\begin{cases}
s_0 \in E & [\, s_0 = \varphi_S(i_0) \,] \\
\text{satisfaisant} & \sigma_0 s_0 = s_0
\end{cases}\]
LaTeX source
\[
(21)\qquad
\begin{cases}
s_0 \in E & [\, s_0 = \varphi_S(i_0) \,] \\
\text{satisfaisant} & \sigma_0 s_0 = s_0
\end{cases}
\]\[(22)\qquad \text{\struck{$s_0 \neq s_1$, $s_0 \in E$}} \quad
s_0,\ s_1 = u s_0,\ s_2 = u^{2} s_0 \quad \text{non alignés.}\]
LaTeX source
\[
(22)\qquad \text{\struck{$s_0 \neq s_1$, $s_0 \in E$}} \quad
s_0,\ s_1 = u s_0,\ s_2 = u^{2} s_0 \quad \text{non alignés.}
\]\[\Pi \simeq \Pi_0(n)\]
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\[ \Pi \simeq \Pi_0(n) \]
\[\sigma_E,\ \sigma'_E \in \mathrm{Aff}(E)\]
LaTeX source
\[
\sigma_E,\ \sigma'_E \in \mathrm{Aff}(E)
\]\[s_0 \in E\]
LaTeX source
\[ s_0 \in E \]
\[\begin{align*}
&(1^{\circ}) && \sigma_E^{2} = \sigma_E'^{2} = \mathrm{id} \\
&(2^{\circ}) && \sigma_E s_0 = s_0 \\
&(3^{\circ}) && s_0,\ s_1 = u_E s_0,\ s_2 = u_E^{2} s_0 \ \text{non alignés}
\end{align*}\]
LaTeX source
\begin{align*}
&(1^{\circ}) && \sigma_E^{2} = \sigma_E'^{2} = \mathrm{id} \\
&(2^{\circ}) && \sigma_E s_0 = s_0 \\
&(3^{\circ}) && s_0,\ s_1 = u_E s_0,\ s_2 = u_E^{2} s_0 \ \text{non alignés}
\end{align*}\[(4^{\circ})\qquad u_E \ \text{est unimodulaire.}\]
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\[
(4^{\circ})\qquad u_E \ \text{est unimodulaire.}
\]\[(23)\qquad \Psi_r(\omega) = \{\, u^{r} \mid u \in \omega \,\}\]
LaTeX source
\[
(23)\qquad \Psi_r(\omega) = \{\, u^{r} \mid u \in \omega \,\}
\]\[\Psi_r(\omega) = \Psi_{r'}(\omega) \iff r' \equiv \pm r \ (n)\]
LaTeX source
\[
\Psi_r(\omega) = \Psi_{r'}(\omega) \iff r' \equiv \pm r \ (n)
\]\[\Psi_r(\Psi_{r'}(\omega)) = \Psi_{rr'}(\omega)\]
LaTeX source
\[
\Psi_r(\Psi_{r'}(\omega)) = \Psi_{rr'}(\omega)
\]\[(24)\qquad (\mathbb{Z}/n\mathbb{Z})^{*}/\{\pm 1\} \overset{\text{déf}}{=} \Psi_n\]
LaTeX source
\[
(24)\qquad (\mathbb{Z}/n\mathbb{Z})^{*}/\{\pm 1\} \overset{\text{déf}}{=} \Psi_n
\]\[\mathbb{Z}_n \xrightarrow{\ r\,\mathrm{id}_{\mathbb{Z}_n}\ } \mathbb{Z}_n\]
LaTeX source
\[
\mathbb{Z}_n \xrightarrow{\ r\,\mathrm{id}_{\mathbb{Z}_n}\ } \mathbb{Z}_n
\]\[u^{r'} = u^{r} \quad \text{ou} \quad u^{r'} = (u^{-1})^{r} \ (= u^{-r})\]
LaTeX source
\[
u^{r'} = u^{r} \quad \text{ou} \quad u^{r'} = (u^{-1})^{r} \ (= u^{-r})
\]\[r' \equiv r \ (n) \quad \text{ou} \quad r' \equiv -r \ (n).\]
LaTeX source
\[
r' \equiv r \ (n) \quad \text{ou} \quad r' \equiv -r \ (n).
\]\[(25)\qquad \Psi_r(u) = u^{r}\]
LaTeX source
\[
(25)\qquad \Psi_r(u) = u^{r}
\]\[(26)\qquad \Psi_r(E) = (E, S, \Psi_r(\omega)) \qquad (\text{si } (r, n) = 1)\]
LaTeX source
\[
(26)\qquad \Psi_r(E) = (E, S, \Psi_r(\omega)) \qquad (\text{si } (r, n) = 1)
\]\[\mathrm{Polreg}_n(E)\]
LaTeX source
\[
\mathrm{Polreg}_n(E)
\]\[(27)\qquad \mathrm{Aut}(\Pi) = \mathrm{Aut}(\Psi_r(\Pi))\]
LaTeX source
\[
(27)\qquad \mathrm{Aut}(\Pi) = \mathrm{Aut}(\Psi_r(\Pi))
\]\[(\mathbb{Z}/p\mathbb{Z})^{*} \simeq \mathbb{Z}/(p-1)\mathbb{Z}\]
LaTeX source
\[
(\mathbb{Z}/p\mathbb{Z})^{*} \simeq \mathbb{Z}/(p-1)\mathbb{Z}
\]\[D_n \xrightarrow{\ \varphi\ } \mathrm{Aff}(E)\]
LaTeX source
\[
D_n \xrightarrow{\ \varphi\ } \mathrm{Aff}(E)
\]\[(28)\qquad
\begin{cases}
\sigma_E,\ u_E \in \mathrm{Aff}(E) \\
\text{satisfaisant} \\
\sigma_E^{2} = \mathrm{id}_E ,\quad \sigma_E u_E \sigma_E^{-1} = u_E^{-1}
\end{cases}\]
LaTeX source
\[
(28)\qquad
\begin{cases}
\sigma_E,\ u_E \in \mathrm{Aff}(E) \\
\text{satisfaisant} \\
\sigma_E^{2} = \mathrm{id}_E ,\quad \sigma_E u_E \sigma_E^{-1} = u_E^{-1}
\end{cases}
\]\[0 \longrightarrow Z \longrightarrow \mathrm{Aff}(S) \longrightarrow (\mathbb{Z}/n\mathbb{Z})^{*} \longrightarrow 0\]
LaTeX source
\[
0 \longrightarrow Z \longrightarrow \mathrm{Aff}(S) \longrightarrow (\mathbb{Z}/n\mathbb{Z})^{*} \longrightarrow 0
\]\[(\mathbb{Z}/n\mathbb{Z})^{*} \simeq \mathrm{Aut}(Z)\]
LaTeX source
\[
(\mathbb{Z}/n\mathbb{Z})^{*} \simeq \mathrm{Aut}(Z)
\]\[Z \subset \mathrm{Aff}(E)\]
LaTeX source
\[ Z \subset \mathrm{Aff}(E) \]\[\begin{cases}
Z_0 = \mu_{n,A} \subset \mathrm{Sl}(2)_{\mathbb{Z}} \subset \mathrm{Aff}\,\mathbb{E}^{2}_{/A} \\
\sigma_0 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \in \mathrm{Sl}(2,\mathbb{Z}) \subset \mathrm{Aff}(\mathbb{E}^{2}_{A}) \\
s_0 = (1,1) \in \mathbb{E}^{2}(A)^{\sigma} \qquad S_0 = Z \cdot s_0
\end{cases}\]
LaTeX source
\[
\begin{cases}
Z_0 = \mu_{n,A} \subset \mathrm{Sl}(2)_{\mathbb{Z}} \subset \mathrm{Aff}\,\mathbb{E}^{2}_{/A} \\
\sigma_0 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \in \mathrm{Sl}(2,\mathbb{Z}) \subset \mathrm{Aff}(\mathbb{E}^{2}_{A}) \\
s_0 = (1,1) \in \mathbb{E}^{2}(A)^{\sigma} \qquad S_0 = Z \cdot s_0
\end{cases}
\]\[\mathbb{D}_n = \{\sigma,u \mid \sigma^{2} = 1,\ \sigma u \sigma^{-1} = u^{-1},\ u^{n} = 1\}
= \{\sigma,\sigma' \mid \sigma^{2} = \sigma'^{2} = 1,\ (\sigma'\sigma)^{n} = 1\}\]
LaTeX source
\[
\mathbb{D}_n = \{\sigma,u \mid \sigma^{2} = 1,\ \sigma u \sigma^{-1} = u^{-1},\ u^{n} = 1\}
= \{\sigma,\sigma' \mid \sigma^{2} = \sigma'^{2} = 1,\ (\sigma'\sigma)^{n} = 1\}
\]\[D_{\infty} = \{\sigma,u \mid \sigma^{2} = 1,\ \sigma u \sigma^{-1} = u^{-1}\}
= \{\sigma,\sigma' \mid \sigma^{2} = \sigma'^{2} = 1\} = \mathbb{Z}_2 * \mathbb{Z}_2\]
LaTeX source
\[
D_{\infty} = \{\sigma,u \mid \sigma^{2} = 1,\ \sigma u \sigma^{-1} = u^{-1}\}
= \{\sigma,\sigma' \mid \sigma^{2} = \sigma'^{2} = 1\} = \mathbb{Z}_2 * \mathbb{Z}_2
\]\[\begin{array}{l} S \subset E \quad (\text{plan affine}) \\ A \subset \mathrm{Dr}(E) \end{array}\]
LaTeX source
\[
\begin{array}{l} S \subset E \quad (\text{plan affine}) \\ A \subset \mathrm{Dr}(E) \end{array}
\]\[\left.\begin{array}{l} S \xrightarrow{\ \varphi_S\ } E \\ A \xrightarrow{\ \varphi_A\ } \mathrm{Dr}(E) \end{array}\right|
\ \text{compatibles avec relations d'incidence}\]
LaTeX source
\[
\left.\begin{array}{l} S \xrightarrow{\ \varphi_S\ } E \\ A \xrightarrow{\ \varphi_A\ } \mathrm{Dr}(E) \end{array}\right|
\ \text{compatibles avec relations d'incidence}
\]\[\varphi_S(s) \text{ incident à } \varphi_A(a) \Longrightarrow s \text{ incident à } a.\]
LaTeX source
\[
\varphi_S(s) \text{ incident à } \varphi_A(a) \Longrightarrow s \text{ incident à } a.
\]\[\mathbb{R}/\mathbb{Z} \simeq U = \{z \in \mathbb{C} \mid z\bar{z} = 1\}.\]
LaTeX source
\[
\mathbb{R}/\mathbb{Z} \simeq U = \{z \in \mathbb{C} \mid z\bar{z} = 1\}.
\]\[\begin{cases}
\zeta^{n} = 1 \qquad \zeta^{d} \neq 1 \text{ si } 1 \leq d < n,\ d \mid n \\
S = \{\zeta^{i} s_0 \mid i \in \mathbb{Z}/n\mathbb{Z}\} \\
a_i = \mathrm{Dr}(s_i, s_{i+1})
\end{cases}\]
LaTeX source
\[
\begin{cases}
\zeta^{n} = 1 \qquad \zeta^{d} \neq 1 \text{ si } 1 \leq d < n,\ d \mid n \\
S = \{\zeta^{i} s_0 \mid i \in \mathbb{Z}/n\mathbb{Z}\} \\
a_i = \mathrm{Dr}(s_i, s_{i+1})
\end{cases}
\]\[\begin{array}{ll}
\sigma_1 & (\text{involution sans pt fixe de } R \text{ telle que } R/\sigma_1 \simeq S) \\
\sigma_0 & (\qquad\text{---}\qquad\text{---}\qquad\text{---}\qquad R/\sigma_0 \simeq A)
\end{array}\]
LaTeX source
\[
\begin{array}{ll}
\sigma_1 & (\text{involution sans pt fixe de } R \text{ telle que } R/\sigma_1 \simeq S) \\
\sigma_0 & (\qquad\text{---}\qquad\text{---}\qquad\text{---}\qquad R/\sigma_0 \simeq A)
\end{array}
\]\[\mathfrak{G}_1 = \{\sigma_0,\sigma_1 \mid \sigma_0^{2} = \sigma_1^{2} = 1\} \simeq \mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/2\mathbb{Z}\]
LaTeX source
\[
\mathfrak{G}_1 = \{\sigma_0,\sigma_1 \mid \sigma_0^{2} = \sigma_1^{2} = 1\} \simeq \mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/2\mathbb{Z}
\]\[(\text{Contours comb.}) \longrightarrow \mathfrak{G}_1\text{-ens.\ ``admissibles''}\]
LaTeX source
\[
(\text{Contours comb.}) \longrightarrow \mathfrak{G}_1\text{-ens.\ ``admissibles''}
\]\[\begin{cases} \sigma = \sigma_1 \\ u = \sigma_1\sigma_0 \end{cases}
\quad\Longrightarrow\quad
\begin{array}{l} \sigma_1 = \sigma \\ \sigma_0 = u^{-1}\sigma \end{array}\]
LaTeX source
\[
\begin{cases} \sigma = \sigma_1 \\ u = \sigma_1\sigma_0 \end{cases}
\quad\Longrightarrow\quad
\begin{array}{l} \sigma_1 = \sigma \\ \sigma_0 = u^{-1}\sigma \end{array}
\]\[\mathfrak{G}_1 = \{\sigma,u \mid \sigma^{2} = 1,\ \sigma u \sigma^{-1} = u^{-1}\}\]
LaTeX source
\[
\mathfrak{G}_1 = \{\sigma,u \mid \sigma^{2} = 1,\ \sigma u \sigma^{-1} = u^{-1}\}
\]\[\mathfrak{G}_1 = \underbrace{\mathbb{Z}\cdot\mathbb{Z}/2\mathbb{Z}}_{\substack{\| \\ D_\infty\ (\text{groupe diédral infini})}}\]
LaTeX source
\[
\mathfrak{G}_1 = \underbrace{\mathbb{Z}\cdot\mathbb{Z}/2\mathbb{Z}}_{\substack{\| \\ D_\infty\ (\text{groupe diédral infini})}}
\]\[\left\{\begin{array}{l} \sigma x \neq x \\ u^{-1}x \neq \sigma x \end{array}\right.
\qquad \forall x \in R\]
LaTeX source
\[
\left\{\begin{array}{l} \sigma x \neq x \\ u^{-1}x \neq \sigma x \end{array}\right.
\qquad \forall x \in R
\]\[u^{-1}x \neq x \qquad \forall x \in R\]
LaTeX source
\[
u^{-1}x \neq x \qquad \forall x \in R
\]\[\forall x \in R, \quad \operatorname{Card}\{x, \sigma x, ux\} = 3\]
LaTeX source
\[
\forall x \in R, \quad \operatorname{Card}\{x, \sigma x, ux\} = 3
\]\[1 \longrightarrow \mathfrak{G}_1^{\circ} \longrightarrow \mathfrak{G}_1 \longrightarrow \mathbb{Z}/2\mathbb{Z} \longrightarrow 1\]
LaTeX source
\[
1 \longrightarrow \mathfrak{G}_1^{\circ} \longrightarrow \mathfrak{G}_1 \longrightarrow \mathbb{Z}/2\mathbb{Z} \longrightarrow 1
\]\[\begin{aligned}
\mathbb{D}_n &= \{\sigma,u \mid \sigma^{2} = 1,\ \sigma u \sigma^{-1} = u^{-1},\ u^{n} = 1\} \\
&= \{\sigma_0,\sigma_1 \mid \sigma_0^{2} = \sigma_1^{2} = (\sigma_0\sigma_1)^{n} = 1\} \\
&\simeq \mathbb{Z}_n\cdot\mathbb{Z}/2\mathbb{Z} \quad (\text{produit semi-direct},\ldots.)
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\mathbb{D}_n &= \{\sigma,u \mid \sigma^{2} = 1,\ \sigma u \sigma^{-1} = u^{-1},\ u^{n} = 1\} \\
&= \{\sigma_0,\sigma_1 \mid \sigma_0^{2} = \sigma_1^{2} = (\sigma_0\sigma_1)^{n} = 1\} \\
&\simeq \mathbb{Z}_n\cdot\mathbb{Z}/2\mathbb{Z} \quad (\text{produit semi-direct},\ldots.)
\end{aligned}
\]\[\begin{cases}
\operatorname{int}(u^{m})(\sigma) = u^{m}\sigma u^{-m} = u^{m}(u^{m}\sigma) = u^{2m}\sigma \\
\operatorname{int}(u^{m})(u\sigma) = u^{2m+1}\sigma
\end{cases}\]
LaTeX source
\[
\begin{cases}
\operatorname{int}(u^{m})(\sigma) = u^{m}\sigma u^{-m} = u^{m}(u^{m}\sigma) = u^{2m}\sigma \\
\operatorname{int}(u^{m})(u\sigma) = u^{2m+1}\sigma
\end{cases}
\]\[\Pi_n = (S_n, A_n, R_n)\]
LaTeX source
\[ \Pi_n = (S_n, A_n, R_n) \]
\[S_n = \langle u\sigma \rangle\backslash \mathbb{D}_n \simeq \mathbb{Z}_n = \{s_i\}_{i \in \mathbb{Z}_n}
\qquad (u\sigma = \sigma_0)\]
LaTeX source
\[
S_n = \langle u\sigma \rangle\backslash \mathbb{D}_n \simeq \mathbb{Z}_n = \{s_i\}_{i \in \mathbb{Z}_n}
\qquad (u\sigma = \sigma_0)
\]\[A_n = \langle\sigma\rangle\backslash \mathbb{D}_n \simeq \mathbb{Z}_n = \{a_i\}_{i \in \mathbb{Z}_n}
\qquad (\sigma = \sigma_1)\]
LaTeX source
\[
A_n = \langle\sigma\rangle\backslash \mathbb{D}_n \simeq \mathbb{Z}_n = \{a_i\}_{i \in \mathbb{Z}_n}
\qquad (\sigma = \sigma_1)
\]\[(s_i, a_i) \quad\text{et}\quad (s_{i+1}, a_i) \quad \text{incidents.}\]
LaTeX source
\[
(s_i, a_i) \quad\text{et}\quad (s_{i+1}, a_i) \quad \text{incidents.}
\]\[\begin{array}{rcl}
\mathrm{Isom}(\Pi,\Pi') & \xrightarrow{\ \sim\ } & \mathrm{Rep}(\Pi') \\
\varphi & \longmapsto & \varphi(r)
\end{array}\]
LaTeX source
\[
\begin{array}{rcl}
\mathrm{Isom}(\Pi,\Pi') & \xrightarrow{\ \sim\ } & \mathrm{Rep}(\Pi') \\
\varphi & \longmapsto & \varphi(r)
\end{array}
\]\[\begin{array}{ll}
\mathrm{Aut}(\Pi_n) \simeq \mathbb{D}_n & (\text{canon.}) \\
\mathrm{Aut}(\Pi) \simeq \mathbb{D}_n & (\text{non canon.}) \text{ si } \Pi \text{ $n$-gone}
\end{array}\]
LaTeX source
\[
\begin{array}{ll}
\mathrm{Aut}(\Pi_n) \simeq \mathbb{D}_n & (\text{canon.}) \\
\mathrm{Aut}(\Pi) \simeq \mathbb{D}_n & (\text{non canon.}) \text{ si } \Pi \text{ $n$-gone}
\end{array}
\]\[G \longrightarrow \mathfrak{S}_{\mathrm{Or}(\Pi)} \simeq \pm 1,\]
LaTeX source
\[
G \longrightarrow \mathfrak{S}_{\mathrm{Or}(\Pi)} \simeq \pm 1,
\]\[1 \longrightarrow G^{+} \longrightarrow G \longrightarrow \{\pm 1\} \longrightarrow 1\]
LaTeX source
\[
1 \longrightarrow G^{+} \longrightarrow G \longrightarrow \{\pm 1\} \longrightarrow 1
\]\[u_{-\omega} = u_\omega^{-1}\]
LaTeX source
\[
u_{-\omega} = u_\omega^{-1}
\]\[\mathrm{Or}(\Pi) \hookrightarrow \mathrm{Génér.}(G^{+})\]
LaTeX source
\[
\mathrm{Or}(\Pi) \hookrightarrow \mathrm{Génér.}(G^{+})
\]\[\Pi \longmapsto \bigl(\mathbf{Z} G^{+},\ \{u_\omega, u_{-\omega}\} \in G^{+},\ S\bigr)\]
LaTeX source
\[
\Pi \longmapsto \bigl(\mathbf{Z} G^{+},\ \{u_\omega, u_{-\omega}\} \in G^{+},\ S\bigr)
\]\[\begin{cases}
\sigma^{2} = \mathrm{id}, \quad \sigma u \sigma^{-1} = u^{-1} \\
u^{n} = 1 \quad (n \text{ \uncertain{minimum}})
\end{cases}\]
LaTeX source
\[
\begin{cases}
\sigma^{2} = \mathrm{id}, \quad \sigma u \sigma^{-1} = u^{-1} \\
u^{n} = 1 \quad (n \text{ \uncertain{minimum}})
\end{cases}
\]\[\zeta\zeta' = 1 \quad\text{ou}\quad \zeta' = \zeta^{-1}.\]
LaTeX source
\[
\zeta\zeta' = 1 \quad\text{ou}\quad \zeta' = \zeta^{-1}.
\]\[u = \begin{pmatrix} \zeta & 0 \\ 0 & \zeta^{-1} \end{pmatrix} \qquad
\sigma = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\]
LaTeX source
\[
u = \begin{pmatrix} \zeta & 0 \\ 0 & \zeta^{-1} \end{pmatrix} \qquad
\sigma = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
\]\[s_0 = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \qquad
s_i = \begin{pmatrix} \zeta^{i} \\ \zeta^{-i} \end{pmatrix}\]
LaTeX source
\[
s_0 = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \qquad
s_i = \begin{pmatrix} \zeta^{i} \\ \zeta^{-i} \end{pmatrix}
\]\[\alpha = \zeta + \zeta' = \zeta + \zeta^{-1} = \operatorname{Tr} u = \operatorname{Tr} u^{-1}\]
LaTeX source
\[
\alpha = \zeta + \zeta' = \zeta + \zeta^{-1} = \operatorname{Tr} u = \operatorname{Tr} u^{-1}
\]\[\{\zeta, \zeta'\} = \{\zeta, \zeta^{-1}\} = \{\zeta', \zeta'^{-1}\}\]
LaTeX source
\[
\{\zeta, \zeta'\} = \{\zeta, \zeta^{-1}\} = \{\zeta', \zeta'^{-1}\}
\]\[\zeta^{2} - \alpha\zeta + 1 = 0\]
LaTeX source
\[
\zeta^{2} - \alpha\zeta + 1 = 0
\]\[\begin{cases}
F(x,y) = G(\sigma_1(x,y), \sigma_2(x,y)) \\
\sigma_1(x,y) = x + y, \quad \sigma_2(x,y) = xy
\end{cases}\]
LaTeX source
\[
\begin{cases}
F(x,y) = G(\sigma_1(x,y), \sigma_2(x,y)) \\
\sigma_1(x,y) = x + y, \quad \sigma_2(x,y) = xy
\end{cases}
\]\[F(x,y) = \sum_{\alpha+\beta = m} c_{\alpha\beta}\, x^{\alpha} y^{\beta}\]
LaTeX source
\[
F(x,y) = \sum_{\alpha+\beta = m} c_{\alpha\beta}\, x^{\alpha} y^{\beta}
\]\[F(u(x,y)) = F(\zeta x, \zeta^{-1} y) = \sum c_{\alpha\beta}\, \zeta^{\alpha-\beta} x^{\alpha} y^{\beta}\]
LaTeX source
\[
F(u(x,y)) = F(\zeta x, \zeta^{-1} y) = \sum c_{\alpha\beta}\, \zeta^{\alpha-\beta} x^{\alpha} y^{\beta}
\]\[F \equiv F \circ u \iff c_{\alpha\beta} = 0 \text{ si } \beta - \alpha \not\equiv 0 \ (n)\]
LaTeX source
\[
F \equiv F \circ u \iff c_{\alpha\beta} = 0 \text{ si } \beta - \alpha \not\equiv 0 \ (n)
\]\[\begin{aligned}
\text{\struck{$\varphi_\alpha(x,y)$}}\qquad
\varphi_{\alpha,i}(x,y) &= x^{\alpha} y^{\alpha+in} + y^{\alpha} x^{\alpha+in} \\
&= x^{\alpha}y^{\alpha}(x^{in} + y^{in}) \\
&= (xy)^{\alpha}\bigl((x^{n})^{i} + (y^{n})^{i}\bigr)
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\text{\struck{$\varphi_\alpha(x,y)$}}\qquad
\varphi_{\alpha,i}(x,y) &= x^{\alpha} y^{\alpha+in} + y^{\alpha} x^{\alpha+in} \\
&= x^{\alpha}y^{\alpha}(x^{in} + y^{in}) \\
&= (xy)^{\alpha}\bigl((x^{n})^{i} + (y^{n})^{i}\bigr)
\end{aligned}
\]\[\alpha \in \mathbf{N}, \quad i \in \mathbf{N}^{*}\]
LaTeX source
\[
\alpha \in \mathbf{N}, \quad i \in \mathbf{N}^{*}
\]\[X^{i} + Y^{i} = S_i(\sigma_1(X,Y), \sigma_2(X,Y))\]
LaTeX source
\[
X^{i} + Y^{i} = S_i(\sigma_1(X,Y), \sigma_2(X,Y))
\]\[= S_i(X+Y, XY)\]
LaTeX source
\[ = S_i(X+Y, XY) \]
\[(x^{n})^{i} + (y^{n})^{i} = S_i(x^{n} + y^{n}, (xy)^{n})\]
LaTeX source
\[
(x^{n})^{i} + (y^{n})^{i} = S_i(x^{n} + y^{n}, (xy)^{n})
\]\[k[x,y]^{D} = k[x,y]^{\Delta_n} = k[\sigma_2, S_n]\]
LaTeX source
\[
k[x,y]^{D} = k[x,y]^{\Delta_n} = k[\sigma_2, S_n]
\]\[\begin{cases}
\sigma_2(x,y) = xy \\
S_n(x,y) = x^{n} + y^{n}
\end{cases}\]
LaTeX source
\[
\begin{cases}
\sigma_2(x,y) = xy \\
S_n(x,y) = x^{n} + y^{n}
\end{cases}
\]\[axy + b \qquad (a, b \in k)\]
LaTeX source
\[ axy + b \qquad (a, b \in k) \]
\[xy - 1 = 0\]
LaTeX source
\[ xy - 1 = 0 \]
\[C \quad\text{d'équation}\quad q_0(x) - 1 = 0\]
LaTeX source
\[
C \quad\text{d'équation}\quad q_0(x) - 1 = 0
\]\[\begin{aligned}
q_0(s_1 - s_0) &= q_0(\zeta - 1, \zeta^{-1} - 1) = (\zeta-1)(\zeta^{-1}-1) \\
&= 2 - \alpha
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
q_0(s_1 - s_0) &= q_0(\zeta - 1, \zeta^{-1} - 1) = (\zeta-1)(\zeta^{-1}-1) \\
&= 2 - \alpha
\end{aligned}
\]\[q_0(a) = 2 - \alpha\]
LaTeX source
\[ q_0(a) = 2 - \alpha \]
\[x^{i}y^{j} \qquad i - j \equiv d \ (n)\]
LaTeX source
\[
x^{i}y^{j} \qquad i - j \equiv d \ (n)
\]\[x^{j+d+nk}\,y^{j} =
\begin{cases}
(xy)^{j}\, x^{d+nk} & \text{si } d + nk \geq 0 \\
(xy)^{\uncertain{j'}}\, y^{-d+nk'} & \text{si } \uncertain{-d+nk' \geq 0}
\end{cases}\]
LaTeX source
\[
x^{j+d+nk}\,y^{j} =
\begin{cases}
(xy)^{j}\, x^{d+nk} & \text{si } d + nk \geq 0 \\
(xy)^{\uncertain{j'}}\, y^{-d+nk'} & \text{si } \uncertain{-d+nk' \geq 0}
\end{cases}
\]\[\operatorname{Sym}^{*}(\check{E})^{G^{+}} \simeq k[xy, x^{n}, y^{n}]
\qquad (U_n V_n - W_n^{n} = 0)\]
LaTeX source
\[
\operatorname{Sym}^{*}(\check{E})^{G^{+}} \simeq k[xy, x^{n}, y^{n}]
\qquad (U_n V_n - W_n^{n} = 0)
\]\[x^{d} = X_d \quad\text{et}\quad y^{n-d} = Y_d,\]
LaTeX source
\[
x^{d} = X_d \quad\text{et}\quad y^{n-d} = Y_d,
\]\[V X_d - (W)^{d} Y_d = 0 \quad \ldots\]
LaTeX source
\[
V X_d - (W)^{d} Y_d = 0 \quad \ldots
\]\[1,\ x,\ y,\ x^{2},\ xy,\ y^{2}\]
LaTeX source
\[
1,\ x,\ y,\ x^{2},\ xy,\ y^{2}
\]\[1,\ \zeta,\ \zeta^{-1},\ \zeta^{2},\ 1,\ \zeta^{-2}\]
LaTeX source
\[
1,\ \zeta,\ \zeta^{-1},\ \zeta^{2},\ 1,\ \zeta^{-2}
\]\[\zeta^{-2},\ \zeta^{-1},\ \zeta^{0} = 1,\ \zeta^{1} = \zeta,\ \zeta^{2}.\]
LaTeX source
\[
\zeta^{-2},\ \zeta^{-1},\ \zeta^{0} = 1,\ \zeta^{1} = \zeta,\ \zeta^{2}.
\]\[\begin{aligned}
&a + bxy && \text{valeur propre } \zeta^{0} = 1 \\
&ax \ (\text{\uncertain{resp.}}\ ay) && \text{valeur propre } \zeta\ (\text{resp.\ } \zeta^{-1}) \\
&ax^{2}\ (\text{resp.\ } ay^{2}) && \text{valeur propre } \zeta^{2}\ (\text{resp.\ } \zeta^{-2})
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
&a + bxy && \text{valeur propre } \zeta^{0} = 1 \\
&ax \ (\text{\uncertain{resp.}}\ ay) && \text{valeur propre } \zeta\ (\text{resp.\ } \zeta^{-1}) \\
&ax^{2}\ (\text{resp.\ } ay^{2}) && \text{valeur propre } \zeta^{2}\ (\text{resp.\ } \zeta^{-2})
\end{aligned}
\]\[\begin{cases}
a + bxy & \text{v.p.\ } 1 \\
ax + by^{2} & \text{v.p.\ } \zeta \\
ay + bx^{2} & \text{v.p.\ } \zeta^{-1}
\end{cases}\]
LaTeX source
\[
\begin{cases}
a + bxy & \text{v.p.\ } 1 \\
ax + by^{2} & \text{v.p.\ } \zeta \\
ay + bx^{2} & \text{v.p.\ } \zeta^{-1}
\end{cases}
\]\[\begin{cases}
a + bxy & \text{v.p.\ } 1 \\
ax \ (\text{\uncertain{resp.}}\ ay) & \text{v.p.\ } \zeta\ (\zeta^{-1}) \\
ax^{2} + by^{2} & \text{v.p.\ } \zeta^{2} = \zeta^{-2}
\end{cases}\]
LaTeX source
\[
\begin{cases}
a + bxy & \text{v.p.\ } 1 \\
ax \ (\text{\uncertain{resp.}}\ ay) & \text{v.p.\ } \zeta\ (\zeta^{-1}) \\
ax^{2} + by^{2} & \text{v.p.\ } \zeta^{2} = \zeta^{-2}
\end{cases}
\]\[(1) \qquad \varphi_G : D_\infty \longrightarrow \operatorname{Aff}(E)\]
LaTeX source
\[
(1) \qquad \varphi_G : D_\infty \longrightarrow \operatorname{Aff}(E)
\]\[(2) \qquad \boxed{\sigma_0, \sigma_1 \in \operatorname{Aff}(E) \qquad \sigma_0^{2} = \sigma_1^{2} = \mathrm{id}_E}\]
LaTeX source
\[
(2) \qquad \boxed{\sigma_0, \sigma_1 \in \operatorname{Aff}(E) \qquad \sigma_0^{2} = \sigma_1^{2} = \mathrm{id}_E}
\]\[(3) \qquad \boxed{r = (s_0, a_0) \in E \times \operatorname{Dr}(E), \quad (s_0, a_0) \text{ incidents}}\]
LaTeX source
\[
(3) \qquad \boxed{r = (s_0, a_0) \in E \times \operatorname{Dr}(E), \quad (s_0, a_0) \text{ incidents}}
\]\[(4) \qquad
\begin{cases}
\varphi_S(\underline{s}_n) = s_n = u^{n} s_0 \\
\varphi_A(\underline{a}_n) = a_n = u^{n} a_0
\end{cases}\]
LaTeX source
\[
(4) \qquad
\begin{cases}
\varphi_S(\underline{s}_n) = s_n = u^{n} s_0 \\
\varphi_A(\underline{a}_n) = a_n = u^{n} a_0
\end{cases}
\]\[(5) \qquad u = \sigma_0 \sigma_1\]
LaTeX source
\[ (5) \qquad u = \sigma_0 \sigma_1 \]
\[\begin{aligned}
&\underline{r} = \underline{g}\,\underline{r}_\infty \qquad (\underline{g} \in \operatorname{Aut}(\Pi_\infty) = D_\infty) \\
(5) \qquad &\varphi_{\mathrm{rep}}(\underline{r}) = \varphi_{\mathrm{rep}}(\underline{g}\,\underline{r}_\infty) = g\,\varphi_{\mathrm{rep}}(\underline{r}_\infty) = g.r
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
&\underline{r} = \underline{g}\,\underline{r}_\infty \qquad (\underline{g} \in \operatorname{Aut}(\Pi_\infty) = D_\infty) \\
(5) \qquad &\varphi_{\mathrm{rep}}(\underline{r}) = \varphi_{\mathrm{rep}}(\underline{g}\,\underline{r}_\infty) = g\,\varphi_{\mathrm{rep}}(\underline{r}_\infty) = g.r
\end{aligned}
\]\[(6) \qquad \varphi(\underline{g}\,\underline{r}_\infty) = g.r \quad\text{si}\quad g = \varphi_G\,\underline{g}\]
LaTeX source
\[
(6) \qquad \varphi(\underline{g}\,\underline{r}_\infty) = g.r \quad\text{si}\quad g = \varphi_G\,\underline{g}
\]\[\sigma_1\, \mathrm{pr}_1(r) = \mathrm{pr}_1(r), \quad \sigma_0\, \mathrm{pr}_2(r) = \mathrm{pr}_2(r)\]
LaTeX source
\[
\sigma_1\, \mathrm{pr}_1(r) = \mathrm{pr}_1(r), \quad \sigma_0\, \mathrm{pr}_2(r) = \mathrm{pr}_2(r)
\]\[(8) \qquad \boxed{\sigma_1 s_0 = s_0, \quad \sigma_0 a_0 = a_0}\]
LaTeX source
\[
(8) \qquad \boxed{\sigma_1 s_0 = s_0, \quad \sigma_0 a_0 = a_0}
\]\[s_0, \quad s_1 = \sigma_0 s_0\ \bigl(= u s_0 = \sigma_0 \underbrace{\sigma_1 s_0}_{s_0}\bigr), \quad
s_{-1} = \sigma_1 s_1\ (= s_{-1} = \sigma_1 \sigma_0 s_0)\]
LaTeX source
\[
s_0, \quad s_1 = \sigma_0 s_0\ \bigl(= u s_0 = \sigma_0 \underbrace{\sigma_1 s_0}_{s_0}\bigr), \quad
s_{-1} = \sigma_1 s_1\ (= s_{-1} = \sigma_1 \sigma_0 s_0)
\]\[(9) \qquad a_0 = \operatorname{dr}(s_0, s_1)\]
LaTeX source
\[
(9) \qquad a_0 = \operatorname{dr}(s_0, s_1)
\]\[\sigma_0 a_0 = \operatorname{dr}(\sigma_0 s_0 = s_1,\ \sigma_1 s_1 = s_{-1})\]
LaTeX source
\[
\sigma_0 a_0 = \operatorname{dr}(\sigma_0 s_0 = s_1,\ \sigma_1 s_1 = s_{-1})
\]\[(10) \qquad
\begin{cases}
\sigma_0, \sigma_1 \in \operatorname{Aff}(E) \\
s_0 \in E
\end{cases}\]
LaTeX source
\[
(10) \qquad
\begin{cases}
\sigma_0, \sigma_1 \in \operatorname{Aff}(E) \\
s_0 \in E
\end{cases}
\]\[(11) \qquad \varphi_G(\underline{\sigma}_0) = \sigma_0, \quad \varphi_G(\underline{\sigma}_1) = \sigma_1\]
LaTeX source
\[
(11) \qquad \varphi_G(\underline{\sigma}_0) = \sigma_0, \quad \varphi_G(\underline{\sigma}_1) = \sigma_1
\]\[\left.
\begin{aligned}
(12) \qquad & \varphi_S(\underline{s}_n) \overset{\mathrm{def}}{=} s_n = u^{n} s_0 \\
(13) \qquad & \varphi_A(\underline{a}_n) = \operatorname{dr}(s_n, s_{n+1}) = u^{n} \operatorname{dr}(s_0, s_1)
\end{aligned}
\right| \; n \in \mathbf{Z}\]
LaTeX source
\[
\left.
\begin{aligned}
(12) \qquad & \varphi_S(\underline{s}_n) \overset{\mathrm{def}}{=} s_n = u^{n} s_0 \\
(13) \qquad & \varphi_A(\underline{a}_n) = \operatorname{dr}(s_n, s_{n+1}) = u^{n} \operatorname{dr}(s_0, s_1)
\end{aligned}
\right| \; n \in \mathbf{Z}
\]\[(14) \qquad u = \sigma_0 \sigma_1\]
LaTeX source
\[ (14) \qquad u = \sigma_0 \sigma_1 \]
\[\begin{cases}
e_1 = s_1 - s_0 \\
e_2 = s_{-1} - s_0
\end{cases}\]
LaTeX source
\[
\begin{cases}
e_1 = s_1 - s_0 \\
e_2 = s_{-1} - s_0
\end{cases}
\]\[(x,y) = s_0 + x e_1 + y e_2 .\]
LaTeX source
\[ (x,y) = s_0 + x e_1 + y e_2 . \]
\[\begin{cases}
s_0 = (0,0), \quad s_1 = (1,0), \quad s_2 = (0,1) \\
e_1 = (1,0) \quad e_2 = (0,1)
\end{cases}\]
LaTeX source
\[
\begin{cases}
s_0 = (0,0), \quad s_1 = (1,0), \quad s_2 = (0,1) \\
e_1 = (1,0) \quad e_2 = (0,1)
\end{cases}
\]\[\begin{cases}
\sigma_0 s_0 = s_1 \\
\sigma_0 s_1 = s_0 \\
\sigma_0 s_{-1} = s_2 = s_0 + (1+\alpha) e_1 + (1+\beta) e_2 \\
\sigma_0 e_1 = -e_1 \\
\sigma_0 e_2 = \alpha e_1 + (1+\beta) e_2
\end{cases}\]
LaTeX source
\[
\begin{cases}
\sigma_0 s_0 = s_1 \\
\sigma_0 s_1 = s_0 \\
\sigma_0 s_{-1} = s_2 = s_0 + (1+\alpha) e_1 + (1+\beta) e_2 \\
\sigma_0 e_1 = -e_1 \\
\sigma_0 e_2 = \alpha e_1 + (1+\beta) e_2
\end{cases}
\]\[\begin{cases}
\beta = 0 \text{ ssi } s_2 - s_{-1} \parallel e_1 \\
\alpha = \zeta + \zeta^{-1} = 2\cos\theta
\end{cases}\]
LaTeX source
\[
\begin{cases}
\beta = 0 \text{ ssi } s_2 - s_{-1} \parallel e_1 \\
\alpha = \zeta + \zeta^{-1} = 2\cos\theta
\end{cases}
\]\[s_2 - s_{-1} = (\zeta^2, \zeta^{-2}) - (\zeta^{-1}, \zeta) = (\zeta^2 - \zeta^{-1}, \zeta^{-2} - \zeta)\]
LaTeX source
\[
s_2 - s_{-1} = (\zeta^2, \zeta^{-2}) - (\zeta^{-1}, \zeta) = (\zeta^2 - \zeta^{-1}, \zeta^{-2} - \zeta)
\]\[s_1 - s_0 = (\zeta - 1, \zeta^{-1} - 1)\]
LaTeX source
\[
s_1 - s_0 = (\zeta - 1, \zeta^{-1} - 1)
\]\[\begin{cases}
(s_2 - s_{-1}) = (1 + \zeta + \zeta^{-1})(s_1 - s_0) = (1+\alpha)(s_1 - s_0) \\
\text{où} \quad \alpha = \zeta + \zeta^{-1}
\end{cases}\]
LaTeX source
\[
\begin{cases}
(s_2 - s_{-1}) = (1 + \zeta + \zeta^{-1})(s_1 - s_0) = (1+\alpha)(s_1 - s_0) \\
\text{où} \quad \alpha = \zeta + \zeta^{-1}
\end{cases}
\]\[\beta = 0 .\]
LaTeX source
\[ \beta = 0 . \]
\[\sigma_0 = \begin{pmatrix} -1 & \alpha & 1 \\ 0 & 1+\beta & 0 \\ 0 & 0 & 1 \end{pmatrix}\]
LaTeX source
\[
\sigma_0 = \begin{pmatrix} -1 & \alpha & 1 \\ 0 & 1+\beta & 0 \\ 0 & 0 & 1 \end{pmatrix}
\]\[\sigma_0(x,y) = (1 - x + \alpha y, (1+\beta) y)\]
LaTeX source
\[ \sigma_0(x,y) = (1 - x + \alpha y, (1+\beta) y) \]
\[\begin{cases}
\sigma_1 s_0 = s_0 \\
\sigma_1 s_1 = s_{-1} \\
\sigma_1 s_{-1} = s_1 \\
\sigma_1 e_1 = e_2 \\
\sigma_1 e_2 = e_1
\end{cases}
\qquad \text{i.e.}\]
LaTeX source
\[
\begin{cases}
\sigma_1 s_0 = s_0 \\
\sigma_1 s_1 = s_{-1} \\
\sigma_1 s_{-1} = s_1 \\
\sigma_1 e_1 = e_2 \\
\sigma_1 e_2 = e_1
\end{cases}
\qquad \text{i.e.}
\]\[\sigma_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}\]
LaTeX source
\[
\sigma_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}
\]\[\sigma_1(x,y) = (y,x)\]
LaTeX source
\[ \sigma_1(x,y) = (y,x) \]
\[\begin{cases}
\boxed{\alpha\beta = 0} \\
(1+\beta)^2 = 1 \quad \text{i.e.} \quad \boxed{\beta^2 + 2\beta = 0} \quad \text{i.e.} \quad \boxed{\beta(\beta+2) = 0}
\end{cases}\]
LaTeX source
\[
\begin{cases}
\boxed{\alpha\beta = 0} \\
(1+\beta)^2 = 1 \quad \text{i.e.} \quad \boxed{\beta^2 + 2\beta = 0} \quad \text{i.e.} \quad \boxed{\beta(\beta+2) = 0}
\end{cases}
\]\[\alpha\beta = (\beta+2)\beta = 0\]
LaTeX source
\[ \alpha\beta = (\beta+2)\beta = 0 \]
\[\begin{cases}
\text{car}\, k = p \neq 2, \quad \sigma_0 = \begin{pmatrix} -1 & 0 & 1 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \\
\sigma_0(x,y) = (1-x, -y)
\end{cases}\]
LaTeX source
\[
\begin{cases}
\text{car}\, k = p \neq 2, \quad \sigma_0 = \begin{pmatrix} -1 & 0 & 1 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \\
\sigma_0(x,y) = (1-x, -y)
\end{cases}
\]\[s_2 - s_{-1} = (1+\alpha) e_1 + \beta e_2\]
LaTeX source
\[
s_2 - s_{-1} = (1+\alpha) e_1 + \beta e_2
\]\[\begin{cases}
\det u_V = -\det \sigma_{0V} = 1 + \beta \\
\operatorname{Tr} \sigma_{0V} = \beta
\end{cases}\]
LaTeX source
\[
\begin{cases}
\det u_V = -\det \sigma_{0V} = 1 + \beta \\
\operatorname{Tr} \sigma_{0V} = \beta
\end{cases}
\]\[\begin{cases}
u(x,y) = (1-y, -x) \\
u^2(x,y) = (x+1, y-1)
\end{cases}\]
LaTeX source
\[
\begin{cases}
u(x,y) = (1-y, -x) \\
u^2(x,y) = (x+1, y-1)
\end{cases}
\]\[\begin{cases}
\sigma_0(x,y) = (1 - x + \alpha y, y) \\
\sigma_1(x,y) = (y,x) \\
u(x,y) = \sigma_0 \sigma_1(x,y) = (1 - y + \alpha x, x)
\end{cases}\]
LaTeX source
\[
\begin{cases}
\sigma_0(x,y) = (1 - x + \alpha y, y) \\
\sigma_1(x,y) = (y,x) \\
u(x,y) = \sigma_0 \sigma_1(x,y) = (1 - y + \alpha x, x)
\end{cases}
\]\[H_0 : \quad 2x - \alpha y - 1 = 0\]
LaTeX source
\[ H_0 : \quad 2x - \alpha y - 1 = 0 \]
\[H_1 : \quad x - y = 0\]
LaTeX source
\[ H_1 : \quad x - y = 0 \]
\[x_0 = y_0 = \frac{1}{2-\alpha}\]
LaTeX source
\[
x_0 = y_0 = \frac{1}{2-\alpha}
\]\[u_V = \begin{pmatrix} \alpha & -1 \\ 1 & 0 \end{pmatrix}\]
LaTeX source
\[
u_V = \begin{pmatrix} \alpha & -1 \\ 1 & 0 \end{pmatrix}
\]\[\begin{cases}
\det u_V = 1 \\
\operatorname{Tr} u_V = \alpha
\end{cases}\]
LaTeX source
\[
\begin{cases}
\det u_V = 1 \\
\operatorname{Tr} u_V = \alpha
\end{cases}
\]\[\zeta^2 - \alpha\zeta + 1 = 0\]
LaTeX source
\[ \zeta^2 - \alpha\zeta + 1 = 0 \]
\[\Delta(\alpha) = \alpha^2 - 4 = (\alpha - 2)(\alpha + 2)\]
LaTeX source
\[ \Delta(\alpha) = \alpha^2 - 4 = (\alpha - 2)(\alpha + 2) \]
\[\alpha = \zeta + \zeta^{-1}\]
LaTeX source
\[
\alpha = \zeta + \zeta^{-1}
\]\[\begin{cases}
\zeta + \zeta' = \alpha \\
\zeta\zeta' = 1
\end{cases}
\quad \text{i.e.} \quad \zeta' = \zeta^{-1} .\]
LaTeX source
\[
\begin{cases}
\zeta + \zeta' = \alpha \\
\zeta\zeta' = 1
\end{cases}
\quad \text{i.e.} \quad \zeta' = \zeta^{-1} .
\]\[\zeta = \zeta' \quad \text{i.e.} \quad \zeta^2 = 1 \quad \text{i.e.} \quad \zeta = \pm 1\]
LaTeX source
\[
\zeta = \zeta' \quad \text{i.e.} \quad \zeta^2 = 1 \quad \text{i.e.} \quad \zeta = \pm 1
\]\[\alpha = \pm 2 \quad \text{ou encore} \quad \Delta(\alpha) = 0\]
LaTeX source
\[
\alpha = \pm 2 \quad \text{ou encore} \quad \Delta(\alpha) = 0
\]\[\begin{array}{l}
\text{\struck{$\sigma_0(x,y) = (1 - x - 2y, y)$}} \\
\text{\struck{$u_0(x,y) = (1 - y - 2x, x)$}} \\
\text{\struck{$u_0^2(x,y) = (1 - x - 2(1 - y - 2x),$}} \\
\text{\struck{$1 - y - 2x) = (-1 + 3x \ldots$}}
\end{array}\]
LaTeX source
\[
\begin{array}{l}
\text{\struck{$\sigma_0(x,y) = (1 - x - 2y, y)$}} \\
\text{\struck{$u_0(x,y) = (1 - y - 2x, x)$}} \\
\text{\struck{$u_0^2(x,y) = (1 - x - 2(1 - y - 2x),$}} \\
\text{\struck{$1 - y - 2x) = (-1 + 3x \ldots$}}
\end{array}
\]\[u_0 = \varepsilon(\mathrm{id} + v_0), \qquad \varepsilon = \pm 1, \quad v_0^2 = 0\]
LaTeX source
\[
u_0 = \varepsilon(\mathrm{id} + v_0), \qquad \varepsilon = \pm 1, \quad v_0^2 = 0
\]\[v_0 = \varepsilon u_0 - \mathrm{id}\]
LaTeX source
\[
v_0 = \varepsilon u_0 - \mathrm{id}
\]\[\begin{array}{l}
u(x,y) = (1 - y + 2\varepsilon x, x) \\
u_0(x,y) = (-y + 2\varepsilon x, x) \\
v_0(x,y) = (x - \varepsilon y, \varepsilon(x - \varepsilon y)) = (x - \varepsilon y)(1, \varepsilon) \\
\phantom{v_0(x,y)} = a'(z)\, a
\end{array}\]
LaTeX source
\[
\begin{array}{l}
u(x,y) = (1 - y + 2\varepsilon x, x) \\
u_0(x,y) = (-y + 2\varepsilon x, x) \\
v_0(x,y) = (x - \varepsilon y, \varepsilon(x - \varepsilon y)) = (x - \varepsilon y)(1, \varepsilon) \\
\phantom{v_0(x,y)} = a'(z)\, a
\end{array}
\]\[\begin{cases}
a'(x,y) = x - \varepsilon y \\
a = (1, \varepsilon)
\end{cases}\]
LaTeX source
\[
\begin{cases}
a'(x,y) = x - \varepsilon y \\
a = (1, \varepsilon)
\end{cases}
\]\[\begin{array}{l}
u^2 = (1 - x + \alpha(1 - y + \alpha x), 1 - y + \alpha x) \\
\phantom{u^2} = ((1+\alpha) + (\alpha^2 - 1)x - \alpha y, 1 + \alpha x - y)
\end{array}\]
LaTeX source
\[
\begin{array}{l}
u^2 = (1 - x + \alpha(1 - y + \alpha x), 1 - y + \alpha x) \\
\phantom{u^2} = ((1+\alpha) + (\alpha^2 - 1)x - \alpha y, 1 + \alpha x - y)
\end{array}
\]\[3 + 3x - 3y, \; 1 + 2x - y\]
LaTeX source
\[ 3 + 3x - 3y, \; 1 + 2x - y \]
\[x = y \qquad 2x = \alpha y \qquad \alpha \neq 2 \qquad
\begin{cases}
\sigma_1(X,Y) = (Y,X) \\
\sigma_0(X,Y) = (-X + \alpha Y, Y) \\
u(X,Y) = (-Y + \alpha X, X)
\end{cases}\]
LaTeX source
\[
x = y \qquad 2x = \alpha y \qquad \alpha \neq 2 \qquad
\begin{cases}
\sigma_1(X,Y) = (Y,X) \\
\sigma_0(X,Y) = (-X + \alpha Y, Y) \\
u(X,Y) = (-Y + \alpha X, X)
\end{cases}
\]\[u = -(1 + v), \qquad v = -1 - u\]
LaTeX source
\[ u = -(1 + v), \qquad v = -1 - u \]
\[\begin{array}{l}
u(X,Y) = (-Y - 2X, X) \\
v(X,Y) = (X + Y, -(X + Y))
\end{array}
\qquad v^2 = 0\]
LaTeX source
\[
\begin{array}{l}
u(X,Y) = (-Y - 2X, X) \\
v(X,Y) = (X + Y, -(X + Y))
\end{array}
\qquad v^2 = 0
\]\[\begin{cases}
u^2 = 1 + 2v = \begin{pmatrix} 1 & \\ 0 & 1 \end{pmatrix} \ldots
\end{cases}\]
LaTeX source
\[
\begin{cases}
u^2 = 1 + 2v = \begin{pmatrix} 1 & \\ 0 & 1 \end{pmatrix} \ldots
\end{cases}
\]\[\begin{cases}
\sigma_1(x,y) = (y,x) \\
\sigma_0(x,y)
\end{cases}
\qquad u_V = \varepsilon(1 + v), \quad \varepsilon = \pm 1\]
LaTeX source
\[
\begin{cases}
\sigma_1(x,y) = (y,x) \\
\sigma_0(x,y)
\end{cases}
\qquad u_V = \varepsilon(1 + v), \quad \varepsilon = \pm 1
\]\[\begin{array}{l}
u(X,Y) = (-Y + 2\varepsilon X, X) = \varepsilon( \ldots \\
u_V = \varepsilon(\mathrm{id} + v) \qquad v = \varepsilon u_V - \mathrm{id}
\end{array}\]
LaTeX source
\[
\begin{array}{l}
u(X,Y) = (-Y + 2\varepsilon X, X) = \varepsilon( \ldots \\
u_V = \varepsilon(\mathrm{id} + v) \qquad v = \varepsilon u_V - \mathrm{id}
\end{array}
\]\[v(X,Y) = (X - \varepsilon Y, \varepsilon(X - \varepsilon Y)) = (X - \varepsilon Y) \cdot (1, \varepsilon)
\qquad X = \varepsilon Y\]
LaTeX source
\[ v(X,Y) = (X - \varepsilon Y, \varepsilon(X - \varepsilon Y)) = (X - \varepsilon Y) \cdot (1, \varepsilon) \qquad X = \varepsilon Y \]
\[\begin{array}{l}
(X + Y, -(X + Y)) \\
(X - Y, -X - Y)
\end{array}\]
LaTeX source
\[
\begin{array}{l}
(X + Y, -(X + Y)) \\
(X - Y, -X - Y)
\end{array}
\]\[\begin{array}{l}
u(x,y) = (1 - y, x) \\
u^2(x,y) = (1 - x, 1 - y) \\
u^4(x,y) = (x, y) \quad \text{---}
\end{array}\]
LaTeX source
\[
\begin{array}{l}
u(x,y) = (1 - y, x) \\
u^2(x,y) = (1 - x, 1 - y) \\
u^4(x,y) = (x, y) \quad \text{---}
\end{array}
\]\[\begin{cases}
X = x - \frac{1}{2-\alpha} \\
Y = y - \frac{1}{2-\alpha}
\end{cases}\]
LaTeX source
\[
\begin{cases}
X = x - \frac{1}{2-\alpha} \\
Y = y - \frac{1}{2-\alpha}
\end{cases}
\]\[\begin{cases}
\sigma_0(X,Y) = (-X + \alpha Y, Y) \\
\sigma_1(X,Y) = (Y, X) \\
u(X,Y) = (-Y + \alpha X, X) = -(\mathrm{id} + a' \otimes a)(X,Y)
\end{cases}\]
LaTeX source
\[
\begin{cases}
\sigma_0(X,Y) = (-X + \alpha Y, Y) \\
\sigma_1(X,Y) = (Y, X) \\
u(X,Y) = (-Y + \alpha X, X) = -(\mathrm{id} + a' \otimes a)(X,Y)
\end{cases}
\]\[a'(X,Y) = X + Y \qquad a = (1, -1)\]
LaTeX source
\[ a'(X,Y) = X + Y \qquad a = (1, -1) \]
\[s_0 = \left(\frac{1}{2-\alpha}, \frac{1}{2-\alpha}\right) = \left(\frac{1}{4}, \frac{1}{4}\right)\]
LaTeX source
\[
s_0 = \left(\frac{1}{2-\alpha}, \frac{1}{2-\alpha}\right) = \left(\frac{1}{4}, \frac{1}{4}\right)
\]\[\begin{array}{l}
u = -(\mathrm{id} + v_0) \\
u^2 = \mathrm{id} + 2 v_0
\end{array}\]
LaTeX source
\[
\begin{array}{l}
u = -(\mathrm{id} + v_0) \\
u^2 = \mathrm{id} + 2 v_0
\end{array}
\]\[\begin{cases}
u^{2n} = \mathrm{id} + 2n v_0 \\
u^{2n+1} = -(\mathrm{id} + (2n+1) v_0)
\end{cases}\]
LaTeX source
\[
\begin{cases}
u^{2n} = \mathrm{id} + 2n v_0 \\
u^{2n+1} = -(\mathrm{id} + (2n+1) v_0)
\end{cases}
\]\[s_{2n} = s_0 + 2n\, \underbrace{v_0(s_0)}_{(\frac12, -\frac12)} = s_0 + n \underbrace{(1,-1)}_{a} = s_0 + na\]
LaTeX source
\[
s_{2n} = s_0 + 2n\, \underbrace{v_0(s_0)}_{(\frac12, -\frac12)} = s_0 + n \underbrace{(1,-1)}_{a} = s_0 + na
\]\[s_{2n+1} = -(s_0 + (2n+1) v_0(s_0)) = -s_0 - \left(n + \tfrac12\right) a\]
LaTeX source
\[
s_{2n+1} = -(s_0 + (2n+1) v_0(s_0)) = -s_0 - \left(n + \tfrac12\right) a
\]\[\phantom{s_{2n+1}} = \left(-s_0 - \tfrac12 a\right) - na\]
LaTeX source
\[
\phantom{s_{2n+1}} = \left(-s_0 - \tfrac12 a\right) - na
\]\[\begin{array}{l}
u_E = \mathrm{id} + w \qquad w^3 = 0 \\
u_E^p = \mathrm{id} + w^p \qquad p \geq 3 \text{ --- } \ldots
\end{array}\]
LaTeX source
\[
\begin{array}{l}
u_E = \mathrm{id} + w \qquad w^3 = 0 \\
u_E^p = \mathrm{id} + w^p \qquad p \geq 3 \text{ --- } \ldots
\end{array}
\]\[f(x,y) = ax^2 + bxy + cy^2 + ux + vy + d\]
LaTeX source
\[ f(x,y) = ax^2 + bxy + cy^2 + ux + vy + d \]
\[f(x,y) = ax^2 + bxy + cy^2 + ux + vy\]
LaTeX source
\[ f(x,y) = ax^2 + bxy + cy^2 + ux + vy \]
\[a = c, \quad u = v \quad \text{i.e.}\]
LaTeX source
\[
a = c, \quad u = v \quad \text{i.e.}
\]\[f(x,y) = a(x^2 + y^2) + bxy + u(x + y)\]
LaTeX source
\[ f(x,y) = a(x^2 + y^2) + bxy + u(x + y) \]
\[f = a f_\alpha\]
LaTeX source
\[ f = a f_\alpha \]
\[\boxed{f_\alpha(x,y) = (x^2 - x) + (y^2 - y) - \alpha xy}\]
LaTeX source
\[
\boxed{f_\alpha(x,y) = (x^2 - x) + (y^2 - y) - \alpha xy}
\]\[f_0(s_i - s_{i-1}) = 1 \qquad \forall i\]
LaTeX source
\[
f_0(s_i - s_{i-1}) = 1 \qquad \forall i
\]\[f_0(X,Y) = X^2 + Y^2 - \alpha XY\]
LaTeX source
\[ f_0(X,Y) = X^2 + Y^2 - \alpha XY \]
\[f_0(1,0) = 1 \; ]\]
LaTeX source
\[ f_0(1,0) = 1 \; ] \]
\[\delta(f_0) = 4 - \alpha^2 = (2 - \alpha)(2 + \alpha)\]
LaTeX source
\[ \delta(f_0) = 4 - \alpha^2 = (2 - \alpha)(2 + \alpha) \]
\[f_0(X,Y) = (Y - \zeta X)(Y - \zeta' X)\]
LaTeX source
\[ f_0(X,Y) = (Y - \zeta X)(Y - \zeta' X) \]
\[Y - \zeta X = 0 \qquad Y - \zeta' X = 0\]
LaTeX source
\[ Y - \zeta X = 0 \qquad Y - \zeta' X = 0 \]
\[(\quad u_0(X,Y) = (-Y + \alpha X, X) \quad )\]
LaTeX source
\[ (\quad u_0(X,Y) = (-Y + \alpha X, X) \quad ) \]
\[X + Y = 0\]
LaTeX source
\[ X + Y = 0 \]
\[\begin{pmatrix} 2 & -\alpha & -1 \\ -\alpha & 2 & -1 \\ -1 & -1 & 0 \end{pmatrix}\]
LaTeX source
\[
\begin{pmatrix} 2 & -\alpha & -1 \\ -\alpha & 2 & -1 \\ -1 & -1 & 0 \end{pmatrix}
\]\[\delta = 2\delta' \qquad \delta' = -(\alpha + 2)\]
LaTeX source
\[ \delta = 2\delta' \qquad \delta' = -(\alpha + 2) \]
\[f(x,y) = (x^2 - x) + (y^2 - y) + 2xy = (x+y)^2 - (x+y)\]
LaTeX source
\[ f(x,y) = (x^2 - x) + (y^2 - y) + 2xy = (x+y)^2 - (x+y) \]
\[= (x+y)(x+y-1) = \lambda(x,y)\,[\lambda(x,y) - 1]\]
LaTeX source
\[ = (x+y)(x+y-1) = \lambda(x,y)\,[\lambda(x,y) - 1] \]
\[x + y = 0, \quad x + y = 1,\]
LaTeX source
\[ x + y = 0, \quad x + y = 1, \]
\[X = x - \tfrac14, \quad Y = y - \tfrac14, \quad x + y = X + Y + \tfrac12,\]
LaTeX source
\[ X = x - \tfrac14, \quad Y = y - \tfrac14, \quad x + y = X + Y + \tfrac12, \]
\[X + Y = -\tfrac12, \quad X + Y = +\tfrac12 \; ).\]
LaTeX source
\[ X + Y = -\tfrac12, \quad X + Y = +\tfrac12 \; ). \]
\[f(x,y) = (x - y)^2 - (x + y) = v^2 - w\]
LaTeX source
\[ f(x,y) = (x - y)^2 - (x + y) = v^2 - w \]
\[\begin{cases}
v = x - y \\
w = x + y
\end{cases}
\quad \text{i.e.} \quad
\begin{cases}
x = \frac12(v + w) \\
y = \frac12(-v + w)
\end{cases}\]
LaTeX source
\[
\begin{cases}
v = x - y \\
w = x + y
\end{cases}
\quad \text{i.e.} \quad
\begin{cases}
x = \frac12(v + w) \\
y = \frac12(-v + w)
\end{cases}
\]\[u(x,y) = (1 - y + 2x, x)\]
LaTeX source
\[ u(x,y) = (1 - y + 2x, x) \]
\[u(v,w) = (v + 1, 1 + 2v + w)\]
LaTeX source
\[ u(v,w) = (v + 1, 1 + 2v + w) \]
\[u(v, v^2) = (v + 1, (v+1)^2) .\]
LaTeX source
\[ u(v, v^2) = (v + 1, (v+1)^2) . \]
\[G \overset{\text{déf}}{=} \operatorname{Im} \underline{G} = \mathrm{Aut}(E, S, A).\]
LaTeX source
\[
G \overset{\text{déf}}{=} \operatorname{Im} \underline{G} = \mathrm{Aut}(E, S, A).
\]\[\underline{v}.s = s \iff v = \mathrm{id} \qquad (v = \varphi_G(\underline{v}) \in \mathrm{Aff}(E)).\]
LaTeX source
\[
\underline{v}.s = s \iff v = \mathrm{id} \qquad (v = \varphi_G(\underline{v}) \in \mathrm{Aff}(E)).
\]\[\begin{array}{l}
\mathbb{Z}[T, T^{-1}] \\
\quad \cup \\
\mathbb{Z}(U) \qquad U = T + T^{-1}
\end{array}\]
LaTeX source
\[
\begin{array}{l}
\mathbb{Z}[T, T^{-1}] \\
\quad \cup \\
\mathbb{Z}(U) \qquad U = T + T^{-1}
\end{array}
\]\[\Sigma_i(U) = T^i + T^{-i} = U^i + \cdots \qquad (i \geq 0)\]
LaTeX source
\[
\Sigma_i(U) = T^i + T^{-i} = U^i + \cdots \qquad (i \geq 0)
\]\[\begin{aligned}
F_{2m}(U) &= (1 + T^2 + \cdots + T^{2(m-1)})\, T^{-(m-1)}\\
&= \Sigma_{m-1}(U) + \Sigma_{m-3}(U) + \cdots = U^{m-1} + \cdots \qquad (m \geq 1)\\
F_{2m+1}(U) &= (1 + T + T^2 + \cdots + T^{2m})\, T^{-m}\\
&= \Sigma_m(U) + \Sigma_{m-1}(U) + \cdots = U^m + \cdots \qquad (m \geq 0)
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
F_{2m}(U) &= (1 + T^2 + \cdots + T^{2(m-1)})\, T^{-(m-1)}\\
&= \Sigma_{m-1}(U) + \Sigma_{m-3}(U) + \cdots = U^{m-1} + \cdots \qquad (m \geq 1)\\
F_{2m+1}(U) &= (1 + T + T^2 + \cdots + T^{2m})\, T^{-m}\\
&= \Sigma_m(U) + \Sigma_{m-1}(U) + \cdots = U^m + \cdots \qquad (m \geq 0)
\end{aligned}
\]\[\left\{
\begin{aligned}
d^\circ(\Sigma_i) &= i\\
d^\circ F_n &= \Bigl[\frac{n-1}{2}\Bigr] = \begin{cases} m-1 & \text{si } n = 2m\\ m & \text{si } n = 2m+1 \end{cases} \qquad n \geq 1
\end{aligned}
\right.\]
LaTeX source
\[
\left\{
\begin{aligned}
d^\circ(\Sigma_i) &= i\\
d^\circ F_n &= \Bigl[\frac{n-1}{2}\Bigr] = \begin{cases} m-1 & \text{si } n = 2m\\ m & \text{si } n = 2m+1 \end{cases} \qquad n \geq 1
\end{aligned}
\right.
\]\[\left\{
\begin{aligned}
F_{2m}(2) &= m, & F_{2m}(-2) &= (-1)^{m-1} m\\
F_{2m+1}(2) &= 2m+1, & F_{2m+1}(-2) &= (-1)^m
\end{aligned}
\right.\]
LaTeX source
\[
\left\{
\begin{aligned}
F_{2m}(2) &= m, & F_{2m}(-2) &= (-1)^{m-1} m\\
F_{2m+1}(2) &= 2m+1, & F_{2m+1}(-2) &= (-1)^m
\end{aligned}
\right.
\]\[\left\{
\begin{aligned}
F_{2m}(U)(T^2 - 1) &= (T^{2m} - 1)\, T^{-(m-1)}\\
F_{2m+1}(U)(T - 1) &= (T^{2m+1} - 1)\, T^{-m}
\end{aligned}
\right.\]
LaTeX source
\[
\left\{
\begin{aligned}
F_{2m}(U)(T^2 - 1) &= (T^{2m} - 1)\, T^{-(m-1)}\\
F_{2m+1}(U)(T - 1) &= (T^{2m+1} - 1)\, T^{-m}
\end{aligned}
\right.
\]\[F_n(U) F_{n+1}(U) (T-1)(T^2-1) = (T^n - 1)(T^{n+1} - 1)\, T^{-(n-1)}\]
LaTeX source
\[
F_n(U) F_{n+1}(U) (T-1)(T^2-1) = (T^n - 1)(T^{n+1} - 1)\, T^{-(n-1)}
\]\[(F_n, F_{n+2}) = (1) \qquad n \geq 1 \quad (\text{et \uncertain{donc} pour } n \in \mathbb{Z})\]
LaTeX source
\[
(F_n, F_{n+2}) = (1) \qquad n \geq 1 \quad (\text{et \uncertain{donc} pour } n \in \mathbb{Z})
\]\[\left.
\begin{aligned}
X'_n &= F_n F_{n+1} = U^{n-1} + \cdots\\
Y'_n &= F_n F_{n-1} = U^{n-2} + \cdots
\end{aligned}
\right| \ (\text{déf})\]
LaTeX source
\[
\left.
\begin{aligned}
X'_n &= F_n F_{n+1} = U^{n-1} + \cdots\\
Y'_n &= F_n F_{n-1} = U^{n-2} + \cdots
\end{aligned}
\right| \ (\text{déf})
\]\[X'_n - U X'_{n-1} + X_{n-2} = 1.\]
LaTeX source
\[
X'_n - U X'_{n-1} + X_{n-2} = 1.
\]\[F_n F_{n+1} - U F_{n-1} F_n + F_{n-2} F_{n-1} = 1\]
LaTeX source
\[
F_n F_{n+1} - U F_{n-1} F_n + F_{n-2} F_{n-1} = 1
\]\[\begin{gathered}
(T^n - 1)(T^{n+1} - 1) - T(T + T^{-1})(T^{n-1} - 1)(T^n - 1)\\
+ T^2 (T^{n-2} - 1)(T^{n-1} - 1) = \text{\struck{\ill{}}}\ (T-1)(T^2-1)\, T^{n-1}
\end{gathered}\]
LaTeX source
\[
\begin{gathered}
(T^n - 1)(T^{n+1} - 1) - T(T + T^{-1})(T^{n-1} - 1)(T^n - 1)\\
+ T^2 (T^{n-2} - 1)(T^{n-1} - 1) = \text{\struck{\ill{}}}\ (T-1)(T^2-1)\, T^{n-1}
\end{gathered}
\]\[\begin{gathered}
T^{2n+1} - T^n - T^{n+1} + 1 - (T^2 + 1)(T^{2n-1} - T^n - T^{n-1} + 1)\\
+ T^{2n-1} - T^n - T^{n+1} + T^2 = \text{\uncertain{id}}
\end{gathered}\]
LaTeX source
\[
\begin{gathered}
T^{2n+1} - T^n - T^{n+1} + 1 - (T^2 + 1)(T^{2n-1} - T^n - T^{n-1} + 1)\\
+ T^{2n-1} - T^n - T^{n+1} + T^2 = \text{\uncertain{id}}
\end{gathered}
\]\[-T^{2n+1} + T^{n+2} + T^{n+1} - T^2 - T^{2n-1} + T^n + T^{n-1} - 1\]
LaTeX source
\[
-T^{2n+1} + T^{n+2} + T^{n+1} - T^2 - T^{2n-1} + T^n + T^{n-1} - 1
\]\[T^{n+2} - T^{n+1} - T^n + T^{n-1} = (T-1)(T^2-1)\, T^{n-1} \quad \text{ok.}\]
LaTeX source
\[
T^{n+2} - T^{n+1} - T^n + T^{n-1} = (T-1)(T^2-1)\, T^{n-1} \quad \text{ok.}
\]\[X'_n = X_n \quad \text{i.e.} \quad X_n = F_n F_{n+1}\]
LaTeX source
\[
X'_n = X_n \quad \text{i.e.} \quad X_n = F_n F_{n+1}
\]\[1 + T^2 + T^4 = (T^{-2} + T^2) + 1 = U^2 - 1\]
LaTeX source
\[
1 + T^2 + T^4 = (T^{-2} + T^2) + 1 = U^2 - 1
\]\[F_{2m+2}(U)(T^2 - 1) = (T^{2(m+1)} - 1)\, T^{-m}\]
LaTeX source
\[
F_{2m+2}(U)(T^2 - 1) = (T^{2(m+1)} - 1)\, T^{-m}
\]\[\left\{
\begin{aligned}
X_n &= F_n F_{n+1}\\
Y_n &= X_{n-1} = F_{n-1} F_n
\end{aligned}
\right. \qquad \Big|\ \forall\, n \in \mathbb{Z}\]
LaTeX source
\[
\left\{
\begin{aligned}
X_n &= F_n F_{n+1}\\
Y_n &= X_{n-1} = F_{n-1} F_n
\end{aligned}
\right. \qquad \Big|\ \forall\, n \in \mathbb{Z}
\]\[\begin{aligned}
T^{n-1} X_n(U) &= A_n(T)\\
T^{n-1} X'_n(U) &= A'_n(T)
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
T^{n-1} X_n(U) &= A_n(T)\\
T^{n-1} X'_n(U) &= A'_n(T)
\end{aligned}
\]\[\text{\struck{\ill{}}}\quad
\left\{\begin{array}{l} F_n(\alpha) F_{n+1}(\alpha)\\ F_{n-1}(\alpha) F_n(\alpha) \end{array}\right.,\quad
\left\{\begin{array}{l} F_{n+1}(\alpha) F_{n+2}(\alpha)\\ F_n(\alpha) F_{n+1}(\alpha) \end{array}\right.,\quad
\left\{\begin{array}{l} F_{n-1}(\alpha) F_n(\alpha)\\ F_{n-2}(\alpha) F_{n-1}(\alpha) \end{array}\right.,\]
LaTeX source
\[
\text{\struck{\ill{}}}\quad
\left\{\begin{array}{l} F_n(\alpha) F_{n+1}(\alpha)\\ F_{n-1}(\alpha) F_n(\alpha) \end{array}\right.,\quad
\left\{\begin{array}{l} F_{n+1}(\alpha) F_{n+2}(\alpha)\\ F_n(\alpha) F_{n+1}(\alpha) \end{array}\right.,\quad
\left\{\begin{array}{l} F_{n-1}(\alpha) F_n(\alpha)\\ F_{n-2}(\alpha) F_{n-1}(\alpha) \end{array}\right.,
\]\[u^n = \begin{pmatrix} F_{n+1}(\alpha)(F_{n+2}(\alpha) - F_n(\alpha)) & & F_n(\alpha) F_{n+1}(\alpha)\\ F_n(\alpha)(F_{n+1}(\alpha) - F_{n-1}(\alpha)) & & F_{n-1}(\alpha) F_n(\alpha)\\ 0 & 0 & 1 \end{pmatrix}\]
LaTeX source
\[
u^n = \begin{pmatrix} F_{n+1}(\alpha)(F_{n+2}(\alpha) - F_n(\alpha)) & & F_n(\alpha) F_{n+1}(\alpha)\\ F_n(\alpha)(F_{n+1}(\alpha) - F_{n-1}(\alpha)) & & F_{n-1}(\alpha) F_n(\alpha)\\ 0 & 0 & 1 \end{pmatrix}
\]\[u^n = \begin{pmatrix}
F_{n+1}(\alpha)(F_{n+2}(\alpha) - F_n(\alpha)) & F_n(\alpha)(F_{n-1}(\alpha) - F_{n+1}(\alpha)) & F_{n+1}(\alpha) F_{n+1}(\alpha)\\
F_n(\alpha)(F_{n+1}(\alpha) - F_{n-1}(\alpha)) & F_{n-1}(\alpha)(F_{n-2}(\alpha) - F_n(\alpha)) & F_{n-1}(\alpha) F_{n+1}(\alpha)\\
0 & 0 & 1
\end{pmatrix}\]
LaTeX source
\[
u^n = \begin{pmatrix}
F_{n+1}(\alpha)(F_{n+2}(\alpha) - F_n(\alpha)) & F_n(\alpha)(F_{n-1}(\alpha) - F_{n+1}(\alpha)) & F_{n+1}(\alpha) F_{n+1}(\alpha)\\
F_n(\alpha)(F_{n+1}(\alpha) - F_{n-1}(\alpha)) & F_{n-1}(\alpha)(F_{n-2}(\alpha) - F_n(\alpha)) & F_{n-1}(\alpha) F_{n+1}(\alpha)\\
0 & 0 & 1
\end{pmatrix}
\]\[u_\alpha(x, y) = (1 - y + \alpha x,\ x)\]
LaTeX source
\[ u_\alpha(x, y) = (1 - y + \alpha x,\ x) \]
\[\begin{aligned}
&\text{\struck{$u_\alpha(s_0)$}}\\
s_1 &= u_\alpha(s_0) = (1, 0)\\
s_{-1} &= u_\alpha^{-1}(s_0) = (0, 1) \quad \text{i.e.}\ u_\alpha(0,1) = (0,0).
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
&\text{\struck{$u_\alpha(s_0)$}}\\
s_1 &= u_\alpha(s_0) = (1, 0)\\
s_{-1} &= u_\alpha^{-1}(s_0) = (0, 1) \quad \text{i.e.}\ u_\alpha(0,1) = (0,0).
\end{aligned}
\]\[\left\{
\begin{aligned}
u_\alpha^n(s_{-1}) &= s_{-1}\\
u_\alpha^n(s_1) &= s_1
\end{aligned}
\right.\]
LaTeX source
\[
\left\{
\begin{aligned}
u_\alpha^n(s_{-1}) &= s_{-1}\\
u_\alpha^n(s_1) &= s_1
\end{aligned}
\right.
\]\[A_n(U) F_{n-1}(U) + B_n(U) F_{n+1}(U) = 1,\]
LaTeX source
\[
A_n(U) F_{n-1}(U) + B_n(U) F_{n+1}(U) = 1,
\]\[\left\{
\begin{aligned}
B_n(U) &= F_n(U),\\
A_n(U) &= F_{n-2}(U) - U F_{n-1}(U)
\end{aligned}
\right.\]
LaTeX source
\[
\left\{
\begin{aligned}
B_n(U) &= F_n(U),\\
A_n(U) &= F_{n-2}(U) - U F_{n-1}(U)
\end{aligned}
\right.
\]\[\Phi_n(T) = \prod_{\substack{\xi \text{ d'ordre } n\\ \text{dans } \overline{\mathbb{Q}}}} (T - \xi)\]
LaTeX source
\[
\Phi_n(T) = \prod_{\substack{\xi \text{ d'ordre } n\\ \text{dans } \overline{\mathbb{Q}}}} (T - \xi)
\]\[\varphi(\underbrace{p_1^{\alpha_1} \cdots p_r^{\alpha_r}}_{n}) = \prod_i \varphi(p_i^{\alpha_i}) = \prod_i \bigl(p_i^{\alpha_i} - p_i^{\alpha_i - 1}\bigr) = n \prod_i \Bigl(1 - \frac{1}{p_i}\Bigr)\]
LaTeX source
\[
\varphi(\underbrace{p_1^{\alpha_1} \cdots p_r^{\alpha_r}}_{n}) = \prod_i \varphi(p_i^{\alpha_i}) = \prod_i \bigl(p_i^{\alpha_i} - p_i^{\alpha_i - 1}\bigr) = n \prod_i \Bigl(1 - \frac{1}{p_i}\Bigr)
\]\[T^n - 1 = \prod_{\substack{d \in \mathbb{N}^*\\ d \mid n}} \Phi_d\]
LaTeX source
\[
T^n - 1 = \prod_{\substack{d \in \mathbb{N}^*\\ d \mid n}} \Phi_d
\]\[\Phi_n = (T^n - 1) \Big/ \prod_{\substack{d \in \mathbb{N}^*\\ d \mid n,\ d \neq n}} \Phi_d\]
LaTeX source
\[
\Phi_n = (T^n - 1) \Big/ \prod_{\substack{d \in \mathbb{N}^*\\ d \mid n,\ d \neq n}} \Phi_d
\]\[\Psi_n(T + T^{-1}) = T^{-\varphi'(n)}\, \Phi_n(T) \qquad \varphi'(n) = \tfrac{1}{2}\varphi(n) \ \text{si } n \geq 3\]
LaTeX source
\[
\Psi_n(T + T^{-1}) = T^{-\varphi'(n)}\, \Phi_n(T) \qquad \varphi'(n) = \tfrac{1}{2}\varphi(n) \ \text{si } n \geq 3
\]\[k'_n = \mathbb{Z}[\tfrac{1}{n}][U] / \Psi_n(U)\]
LaTeX source
\[
k'_n = \mathbb{Z}[\tfrac{1}{n}][U] / \Psi_n(U)
\]\[\Phi_p(T) = \frac{T^p - 1}{T - 1} = T^{p-1} + \cdots + 1\]
LaTeX source
\[
\Phi_p(T) = \frac{T^p - 1}{T - 1} = T^{p-1} + \cdots + 1
\]\[\Psi_p(U) = F_p(U)\]
LaTeX source
\[ \Psi_p(U) = F_p(U) \]
\[k'_p = \mathbb{Z}[U] / \Psi_p(U) = \mathbb{Z}[U] / F_p(U)\]
LaTeX source
\[
k'_p = \mathbb{Z}[U] / \Psi_p(U) = \mathbb{Z}[U] / F_p(U)
\]\[\xi^2 \neq 1,\ \xi^p \neq 1 \quad \text{sont \uncertain{aussi}} \quad (\xi^2)^p = 1,\ \xi^2 \neq 1\]
LaTeX source
\[
\xi^2 \neq 1,\ \xi^p \neq 1 \quad \text{sont \uncertain{aussi}} \quad (\xi^2)^p = 1,\ \xi^2 \neq 1
\]\[\Phi_{2p}(T) = \Phi_p(-T) = T^{p-1} - T^{p-2} + \cdots + 1\]
LaTeX source
\[
\Phi_{2p}(T) = \Phi_p(-T) = T^{p-1} - T^{p-2} + \cdots + 1
\]\[\text{\struck{\ill{}}}\quad \Psi_{2p}(U) = \Psi_p(-U)\,(-1)^{\frac{p-1}{2}}\]
LaTeX source
\[
\text{\struck{\ill{}}}\quad \Psi_{2p}(U) = \Psi_p(-U)\,(-1)^{\frac{p-1}{2}}
\]\[k'_{2p} = \mathbb{Z}[\tfrac{1}{2}][U] / \Psi_{2p}(U).\]
LaTeX source
\[
k'_{2p} = \mathbb{Z}[\tfrac{1}{2}][U] / \Psi_{2p}(U).
\]\[k'_4 = \mathbb{Z}[U] / U \simeq \mathbb{Z}\]
LaTeX source
\[
k'_4 = \mathbb{Z}[U] / U \simeq \mathbb{Z}
\]\[\Phi_n(T) \equiv \Phi_{n'}(T)^{\varphi(p^r)} \mod p\]
LaTeX source
\[
\Phi_n(T) \equiv \Phi_{n'}(T)^{\varphi(p^r)} \mod p
\]\[(1) \qquad \Phi_\delta(T) = \Phi_{p^s d}(T) \equiv \Phi_d(T)^{\varphi(p^s)}\]
LaTeX source
\[
(1) \qquad \Phi_\delta(T) = \Phi_{p^s d}(T) \equiv \Phi_d(T)^{\varphi(p^s)}
\]\[(2) \qquad \text{\struck{$\Phi_{n'}(T)^{\varphi(p^r)}$}}\ \Phi_n(T) = \Phi_{p^r n'}(T) \equiv \text{\struck{$\Phi_{n'}(T)^{\varphi(p^r)}\, \lambda(T)$}}\ \text{\add{$[\Phi_n(T)]$}}\]
LaTeX source
\[
(2) \qquad \text{\struck{$\Phi_{n'}(T)^{\varphi(p^r)}$}}\ \Phi_n(T) = \Phi_{p^r n'}(T) \equiv \text{\struck{$\Phi_{n'}(T)^{\varphi(p^r)}\, \lambda(T)$}}\ \text{\add{$[\Phi_n(T)]$}}
\]\[\text{\struck{où $\lambda(T) = \dfrac{\Phi_n(T)}{\Phi_{n'}(T)^{\varphi(p^r)}}$}}\]
LaTeX source
\[
\text{\struck{où $\lambda(T) = \dfrac{\Phi_n(T)}{\Phi_{n'}(T)^{\varphi(p^r)}}$}}
\]\[\Phi_{n'}(T)^{\varphi(p^r)} \underbrace{\prod_{\delta \mid n} \Phi_\delta(T)}_{(T^n - 1)}
\equiv
\underbrace{\Bigl(\prod_{\substack{0 \leq s \leq r\\ 1 \leq d \mid n'}} \Phi_d(T)^{\varphi(p^s)}\Bigr)}_{\prod_{0 \leq s \leq r} \bigl(\prod_{1 \leq d \mid n'} \Phi_d(T)\bigr)^{\varphi(p^s)}} \Phi_n(T)\]
LaTeX source
\[
\Phi_{n'}(T)^{\varphi(p^r)} \underbrace{\prod_{\delta \mid n} \Phi_\delta(T)}_{(T^n - 1)}
\equiv
\underbrace{\Bigl(\prod_{\substack{0 \leq s \leq r\\ 1 \leq d \mid n'}} \Phi_d(T)^{\varphi(p^s)}\Bigr)}_{\prod_{0 \leq s \leq r} \bigl(\prod_{1 \leq d \mid n'} \Phi_d(T)\bigr)^{\varphi(p^s)}} \Phi_n(T)
\]\[\Bigl(\prod_{1 \leq d \mid n'} \Phi_d(T)\Bigr)^{\varphi(p^s)} = (T^{n'} - 1)^{\varphi(p^s)}\]
LaTeX source
\[
\Bigl(\prod_{1 \leq d \mid n'} \Phi_d(T)\Bigr)^{\varphi(p^s)} = (T^{n'} - 1)^{\varphi(p^s)}
\]\[\Phi_{n'}(T)^{\varphi(p^r)} (T^n - 1) \equiv \text{\struck{\ill{}}}\ (T^{n'} - 1)^{\sum_{s=0}^{r} \varphi(p^s)}\, \Phi_n(T)\]
LaTeX source
\[
\Phi_{n'}(T)^{\varphi(p^r)} (T^n - 1) \equiv \text{\struck{\ill{}}}\ (T^{n'} - 1)^{\sum_{s=0}^{r} \varphi(p^s)}\, \Phi_n(T)
\]\[\sum_{s=0}^{r} \varphi(p^s) = 1 + (p - 1) + (p^2 - p) + \cdots + (p^r - p^{r-1}) = p^r,\]
LaTeX source
\[
\sum_{s=0}^{r} \varphi(p^s) = 1 + (p - 1) + (p^2 - p) + \cdots + (p^r - p^{r-1}) = p^r,
\]\[\Phi_{n'}(T)^{\varphi(p^r)} (T^n - 1) \equiv (T^{n'} - 1)^{p^r}\, \Phi_n(T)\]
LaTeX source
\[
\Phi_{n'}(T)^{\varphi(p^r)} (T^n - 1) \equiv (T^{n'} - 1)^{p^r}\, \Phi_n(T)
\]\[T^n - 1 = (T^{n'})^{p^r} - (1)^{p^r} \equiv (T^{n'} - 1)^{p^r}\]
LaTeX source
\[
T^n - 1 = (T^{n'})^{p^r} - (1)^{p^r} \equiv (T^{n'} - 1)^{p^r}
\]\[\Phi_{n'}(T)^{\varphi(p^r)} = \Phi_n(T)\]
LaTeX source
\[
\Phi_{n'}(T)^{\varphi(p^r)} = \Phi_n(T)
\]\[\Psi_n(\alpha) = 0\]
LaTeX source
\[ \Psi_n(\alpha) = 0 \]
\[n = p^r n', \qquad (p, n') = 1.\]
LaTeX source
\[ n = p^r n', \qquad (p, n') = 1. \]
\[\begin{array}{r|c|c|c||c||c}
p^r = & 2 & 4 & 8 & 3 & 5\\ \hline
\varphi(p^r) = & 1 & 2 & 4 & 2 & 4
\end{array}\]
LaTeX source
\[
\begin{array}{r|c|c|c||c||c}
p^r = & 2 & 4 & 8 & 3 & 5\\ \hline
\varphi(p^r) = & 1 & 2 & 4 & 2 & 4
\end{array}
\]\[\left.
\begin{array}{ll}
\text{Cas } \varphi(n) = 1 & n = 1, 2\\
\bigl[\text{si } n \geq 3,\ \varphi(n) \text{ est \textit{pair}}\bigr] & \\
\text{Cas } \varphi(n) = 2 & n = 4, 3, 6
\end{array}
\right\} \ \text{Cas } \alpha \text{ entier}\]
LaTeX source
\[
\left.
\begin{array}{ll}
\text{Cas } \varphi(n) = 1 & n = 1, 2\\
\bigl[\text{si } n \geq 3,\ \varphi(n) \text{ est \textit{pair}}\bigr] & \\
\text{Cas } \varphi(n) = 2 & n = 4, 3, 6
\end{array}
\right\} \ \text{Cas } \alpha \text{ entier}
\]\[\left.
\text{Cas } \varphi(n) = 4 \quad
\left\{
\begin{array}{ll}
n = p^r & n = 8,\ n = 5\\
n = p^r q^s & 2.5 = 10 \qquad 4.3 = 12
\end{array}
\right.
\right\} \ \text{Cas } \alpha \text{ quadratiques}\]
LaTeX source
\[
\left.
\text{Cas } \varphi(n) = 4 \quad
\left\{
\begin{array}{ll}
n = p^r & n = 8,\ n = 5\\
n = p^r q^s & 2.5 = 10 \qquad 4.3 = 12
\end{array}
\right.
\right\} \ \text{Cas } \alpha \text{ quadratiques}
\]\[\begin{array}{c|c|c|c||c|c||}
n & 1 & 2 & 4 & 3 & 6\\ \hline
\alpha & 2 & -2 & 0 & -1 & +1
\end{array}\]
LaTeX source
\[
\begin{array}{c|c|c|c||c|c||}
n & 1 & 2 & 4 & 3 & 6\\ \hline
\alpha & 2 & -2 & 0 & -1 & +1
\end{array}
\]\[\begin{array}{c|c||c||c|c||}
n & 8 & 12 & 5 & 10\\ \hline
\alpha & \pm\sqrt{2} & \pm\sqrt{3} & \dfrac{-1 \pm \sqrt{5}}{2} & \dfrac{1 \pm \sqrt{5}}{2}\\
& \alpha^2 - 2 = 0 & \alpha^2 - 3 = 0 & \alpha^2 + \alpha - 1 = 0 & \alpha^2 - \alpha - 1 = 0
\end{array}\]
LaTeX source
\[
\begin{array}{c|c||c||c|c||}
n & 8 & 12 & 5 & 10\\ \hline
\alpha & \pm\sqrt{2} & \pm\sqrt{3} & \dfrac{-1 \pm \sqrt{5}}{2} & \dfrac{1 \pm \sqrt{5}}{2}\\
& \alpha^2 - 2 = 0 & \alpha^2 - 3 = 0 & \alpha^2 + \alpha - 1 = 0 & \alpha^2 - \alpha - 1 = 0
\end{array}
\]\[\begin{aligned}
n &= p^r & \varphi(n) &= p^r - p^{r-1} = 6\\
n &= p^r q^s & &\{p^r = 2,\ q^s = 9, 7\}
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
n &= p^r & \varphi(n) &= p^r - p^{r-1} = 6\\
n &= p^r q^s & &\{p^r = 2,\ q^s = 9, 7\}
\end{aligned}
\]\[\begin{array}{c|c|c|}
n & 9 & 7\\ \hline
\alpha & \alpha^3 - 3\alpha + 1 = 0 & \alpha^3 + \alpha^2 - 2\alpha - 1 = 0
\end{array}\]
LaTeX source
\[
\begin{array}{c|c|c|}
n & 9 & 7\\ \hline
\alpha & \alpha^3 - 3\alpha + 1 = 0 & \alpha^3 + \alpha^2 - 2\alpha - 1 = 0
\end{array}
\]\[\begin{array}{c|c|c|}
n & 18 & 14\\ \hline
\alpha & \alpha^3 - 3\alpha - 1 = 0 & \alpha^3 - \alpha^2 - 2\alpha + 1 = 0
\end{array}\]
LaTeX source
\[
\begin{array}{c|c|c|}
n & 18 & 14\\ \hline
\alpha & \alpha^3 - 3\alpha - 1 = 0 & \alpha^3 - \alpha^2 - 2\alpha + 1 = 0
\end{array}
\]\[({}^t u\, x')\otimes x = x'\otimes u x\]
LaTeX source
\[
({}^t u\, x')\otimes x = x'\otimes u x
\]\[\bigl[{}^t u(-2e'_1+e'_2)\bigr]\otimes e_1 = (-2e'_1+e'_2)\otimes u e_1\]
LaTeX source
\[
\bigl[{}^t u(-2e'_1+e'_2)\bigr]\otimes e_1 = (-2e'_1+e'_2)\otimes u e_1
\]\[{}^t u(e'_1) = \sum u_{1i}\, e'_i, \qquad {}^t u\, e'_2 = \sum u_{2i}\, e'_i, \qquad u e_1 = \sum_i u_{i1}\, e_i\]
LaTeX source
\[
{}^t u(e'_1) = \sum u_{1i}\, e'_i, \qquad {}^t u\, e'_2 = \sum u_{2i}\, e'_i, \qquad u e_1 = \sum_i u_{i1}\, e_i
\]\[= \sum(-2u_{i1})\, e'_1\otimes e_i + \sum u_{i1}\, e'_2\otimes e_i\]
LaTeX source
\[
= \sum(-2u_{i1})\, e'_1\otimes e_i + \sum u_{i1}\, e'_2\otimes e_i
\]\[\begin{array}{cccc}
e'_1\otimes e_1 & e'_2\otimes e_1 & \cdots & e'_n\otimes e_1\\
e'_1\otimes e_2 & e'_2\otimes e_2 & & \\
\vdots & \vdots & & \\
e'_1\otimes e_n & e'_2\otimes e_n & &
\end{array}\]
LaTeX source
\[
\begin{array}{cccc}
e'_1\otimes e_1 & e'_2\otimes e_1 & \cdots & e'_n\otimes e_1\\
e'_1\otimes e_2 & e'_2\otimes e_2 & & \\
\vdots & \vdots & & \\
e'_1\otimes e_n & e'_2\otimes e_n & &
\end{array}
\]\[\begin{cases} u_{i1} = 0 \\ -2u_{i1} = 0 \end{cases} \quad i\geqslant 2
\qquad\qquad
u_{2i} = 2u_{1i} \quad i\geqslant 3\]
LaTeX source
\[
\begin{cases} u_{i1} = 0 \\ -2u_{i1} = 0 \end{cases} \quad i\geqslant 2
\qquad\qquad
u_{2i} = 2u_{1i} \quad i\geqslant 3
\]\[-2u_{11}+u_{21} = -2u_{11}, \qquad -2u_{12}+u_{22} = u_{11}\]
LaTeX source
\[
-2u_{11}+u_{21} = -2u_{11}, \qquad -2u_{12}+u_{22} = u_{11}
\]\[\begin{array}{ccccc}
\lambda & u_{12} & u_{13} & \cdots & u_{1n}\\
0 & \lambda & 2u_{13} & \cdots & 2u_{1n}\\
\vdots & & & & \\
0 & & & &
\end{array}\]
LaTeX source
\[
\begin{array}{ccccc}
\lambda & u_{12} & u_{13} & \cdots & u_{1n}\\
0 & \lambda & 2u_{13} & \cdots & 2u_{1n}\\
\vdots & & & & \\
0 & & & &
\end{array}
\]\[\lambda\,(T-1)^{\nu-1}(T+1)\]
LaTeX source
\[
\lambda\,(T-1)^{\nu-1}(T+1)
\]\[(\lambda-1)^{\nu-1}(\lambda+1) = 0\]
LaTeX source
\[
(\lambda-1)^{\nu-1}(\lambda+1) = 0
\]\[\Phi_n(T) = T^{\varphi'(n)}\, F_n(T+T^{-1}) \qquad (\varphi'(n)=\varphi(n)/2)\]
LaTeX source
\[
\Phi_n(T) = T^{\varphi'(n)}\, F_n(T+T^{-1}) \qquad (\varphi'(n)=\varphi(n)/2)
\]\[\deg F_n = \begin{cases} 1 & \text{si } n=1,2 \quad (F_1(T)=T-2,\ F_2(T)=T+2)\\ \varphi(n)/2 & \text{si } n\geqslant 3\end{cases}\]
LaTeX source
\[
\deg F_n = \begin{cases} 1 & \text{si } n=1,2 \quad (F_1(T)=T-2,\ F_2(T)=T+2)\\ \varphi(n)/2 & \text{si } n\geqslant 3\end{cases}
\]\[\prod_{\substack{d\in\mathbb{N}^*\\ d\mid n}} \Phi_d(T) = T^n-1\]
LaTeX source
\[
\prod_{\substack{d\in\mathbb{N}^*\\ d\mid n}} \Phi_d(T) = T^n-1
\]\[\begin{array}{ll}
F_1(U) = U-2 & \Phi_1(T) = T-1\\
F_2(U) = U+2 & \Phi_2(T) = T+1\\
F_3(U) = U+1 & \Phi_3(T) = T^2+T+1\\
F_4(U) = U & \Phi_4(T) = T^2+1\\
F_5(U) = U^2+U-1 & \Phi_5(T) = T^4+T^3+T^2+T+1\\
F_6(U) = U-1 & \Phi_6(T) = T^2-T+1\\
F_7(U) = U^3+U^2-2U-1 & \Phi_7(T) = T^6+T^5+T^4+T^3+T^2+T+1\\
F_8(U) = U^2-2 & \Phi_8(T) = T^4+1\\
F_9(U) = U^3-3U+1 & \Phi_9(T) = T^6+T^3+1\\
F_{10}(U) = U^2-U-1 & \Phi_{10}(T) = T^4-T^3+T^2-T+1\\
\cdots & \cdots
\end{array}\]
LaTeX source
\[
\begin{array}{ll}
F_1(U) = U-2 & \Phi_1(T) = T-1\\
F_2(U) = U+2 & \Phi_2(T) = T+1\\
F_3(U) = U+1 & \Phi_3(T) = T^2+T+1\\
F_4(U) = U & \Phi_4(T) = T^2+1\\
F_5(U) = U^2+U-1 & \Phi_5(T) = T^4+T^3+T^2+T+1\\
F_6(U) = U-1 & \Phi_6(T) = T^2-T+1\\
F_7(U) = U^3+U^2-2U-1 & \Phi_7(T) = T^6+T^5+T^4+T^3+T^2+T+1\\
F_8(U) = U^2-2 & \Phi_8(T) = T^4+1\\
F_9(U) = U^3-3U+1 & \Phi_9(T) = T^6+T^3+1\\
F_{10}(U) = U^2-U-1 & \Phi_{10}(T) = T^4-T^3+T^2-T+1\\
\cdots & \cdots
\end{array}
\]\[\begin{cases}
\Phi_p(T) = 1+T+\cdots+T^{p-1} \quad \text{si } p \text{ premier}\\
F_p(T) = S_0(T)+S_1(T)+\cdots+S_{p'}(T)
\end{cases}\]
LaTeX source
\[
\begin{cases}
\Phi_p(T) = 1+T+\cdots+T^{p-1} \quad \text{si } p \text{ premier}\\
F_p(T) = S_0(T)+S_1(T)+\cdots+S_{p'}(T)
\end{cases}
\]\[\begin{cases} S_0(U) = 1\\ T^n+T^{-n} = S_n(T+T^{-1}) & n\geqslant 1\end{cases}\]
LaTeX source
\[
\begin{cases} S_0(U) = 1\\ T^n+T^{-n} = S_n(T+T^{-1}) & n\geqslant 1\end{cases}
\]\[\begin{cases} S_0(U)=1\\ S_1(U) = U\\ S_2(U) = U^2-2\\ S_3(U) = U^3-3U\\ \cdots\end{cases}\]
LaTeX source
\[
\begin{cases} S_0(U)=1\\ S_1(U) = U\\ S_2(U) = U^2-2\\ S_3(U) = U^3-3U\\ \cdots\end{cases}
\]\[U^n = \sum_{0\leqslant i\leqslant[\frac n2]} \binom{n}{i} S_{n-2i}(U) \qquad n\geqslant 1\]
LaTeX source
\[
U^n = \sum_{0\leqslant i\leqslant[\frac n2]} \binom{n}{i} S_{n-2i}(U) \qquad n\geqslant 1
\]\[S_n(U) = U^n - \sum_{1\leqslant i\leqslant[\frac n2]}\binom{n}{i} S_{n-2i}(U)\]
LaTeX source
\[
S_n(U) = U^n - \sum_{1\leqslant i\leqslant[\frac n2]}\binom{n}{i} S_{n-2i}(U)
\]\[\begin{cases}
\Phi_{2p}(T) = 1-T+T^2+\cdots+(-1)^{p-1}T^{p-1}\\
F_{2p}(T) = (-1)^{p'}\,\Phi_p(-T)\\
\phantom{F_{2p}(T)} = \bigl(S_0(T)-S_1(T)+\cdots+(-1)^{p'}S_{p'}(T)\bigr)(-1)^{p'}
\end{cases}\]
LaTeX source
\[
\begin{cases}
\Phi_{2p}(T) = 1-T+T^2+\cdots+(-1)^{p-1}T^{p-1}\\
F_{2p}(T) = (-1)^{p'}\,\Phi_p(-T)\\
\phantom{F_{2p}(T)} = \bigl(S_0(T)-S_1(T)+\cdots+(-1)^{p'}S_{p'}(T)\bigr)(-1)^{p'}
\end{cases}
\]\[\begin{cases}
\text{multiplicité } 1 & \text{si } (p,n)=1\\
\text{multiplicité } \varphi(p^r) & \text{si } n=p^r n',\ (n,n')=1,\ r\in\mathbb{N}
\end{cases}\]
LaTeX source
\[
\begin{cases}
\text{multiplicité } 1 & \text{si } (p,n)=1\\
\text{multiplicité } \varphi(p^r) & \text{si } n=p^r n',\ (n,n')=1,\ r\in\mathbb{N}
\end{cases}
\]\[\Phi_n(T) = \Bigl(\prod(T-\zeta_i)\Bigr)^{\varphi(p^r)} = \Phi_{n'}(T)^{\varphi(p^r)} \quad\text{dans } k[T]\]
LaTeX source
\[
\Phi_n(T) = \Bigl(\prod(T-\zeta_i)\Bigr)^{\varphi(p^r)} = \Phi_{n'}(T)^{\varphi(p^r)} \quad\text{dans } k[T]
\]\[\Phi_n(T) = \Phi_{n'}(T)^{\varphi(p^r)} \quad\text{dans } \mathbb{F}_p(T)\]
LaTeX source
\[
\Phi_n(T) = \Phi_{n'}(T)^{\varphi(p^r)} \quad\text{dans } \mathbb{F}_p(T)
\]\[\Phi_n \equiv \Phi_{n'}^{\varphi(p^r)} \mod p\]
LaTeX source
\[
\Phi_n \equiv \Phi_{n'}^{\varphi(p^r)} \mod p
\]\[\begin{cases}
\Phi_{p^r}(T) \equiv (T-1)^{\varphi(p^r)} \mod p\\
\Phi_p(T) \equiv (T-1)^{p-1} \mod p
\end{cases}\]
LaTeX source
\[
\begin{cases}
\Phi_{p^r}(T) \equiv (T-1)^{\varphi(p^r)} \mod p\\
\Phi_p(T) \equiv (T-1)^{p-1} \mod p
\end{cases}
\]\[\begin{cases}
F_n(T) \equiv F_{n'}(T)^{\varphi(p^r)} \mod p & \text{si } n'\neq 1,2\\
F_n(T) \equiv F_{n'}(T)^{\varphi(p^r)/2} \mod p & \text{si } n'=1 \text{ ou } 2
\end{cases}\]
LaTeX source
\[
\begin{cases}
F_n(T) \equiv F_{n'}(T)^{\varphi(p^r)} \mod p & \text{si } n'\neq 1,2\\
F_n(T) \equiv F_{n'}(T)^{\varphi(p^r)/2} \mod p & \text{si } n'=1 \text{ ou } 2
\end{cases}
\]\[\begin{cases}
F_{p^r}(T) \equiv \text{\struck{$F_p(T)^{\varphi(p)}$}}\ (T-2)^{\varphi(p^r)} \quad (p)\\
F_p(T) \equiv (T-2)
\end{cases}\]
LaTeX source
\[
\begin{cases}
F_{p^r}(T) \equiv \text{\struck{$F_p(T)^{\varphi(p)}$}}\ (T-2)^{\varphi(p^r)} \quad (p)\\
F_p(T) \equiv (T-2)
\end{cases}
\]\[n = n'p^r \qquad ((n,n')=1,\ r\in\mathbb{N})\]
LaTeX source
\[
n = n'p^r \qquad ((n,n')=1,\ r\in\mathbb{N})
\]\[F_n(T) \equiv F_{n'}(T)^{\varphi(p^r)} \mod p\]
LaTeX source
\[
F_n(T) \equiv F_{n'}(T)^{\varphi(p^r)} \mod p
\]\[F_n(T) \equiv F_{n'}(T)^{\varphi(p^r)/2} \mod (p)
\quad
\begin{cases}
(T-2)^{\varphi(p^r)/2} & \text{si } n=p^r\\
(T+2)^{\varphi(p^r)/2} & \text{si } n=2p^r
\end{cases}\]
LaTeX source
\[
F_n(T) \equiv F_{n'}(T)^{\varphi(p^r)/2} \mod (p)
\quad
\begin{cases}
(T-2)^{\varphi(p^r)/2} & \text{si } n=p^r\\
(T+2)^{\varphi(p^r)/2} & \text{si } n=2p^r
\end{cases}
\]\[\begin{cases}
F_p(T) \equiv (T-1)^{p'} \mod p\\
F_{2p}(T) \equiv (T+1)^{p'} \mod p
\end{cases}
\qquad \Big|\ p' = (p-1)/2\]
LaTeX source
\[
\begin{cases}
F_p(T) \equiv (T-1)^{p'} \mod p\\
F_{2p}(T) \equiv (T+1)^{p'} \mod p
\end{cases}
\qquad \Big|\ p' = (p-1)/2
\]\[\begin{array}{l}
\alpha\in B_{n(p)} = \mathbb{Z}[U]/F_p(U)\\
\zeta\in A_p = \mathbb{Z}[T]/\Phi_p(T)
\end{array}
\quad\Big|\ \text{normaux} \qquad \Big|\ p \text{ premier impair}\]
LaTeX source
\[
\begin{array}{l}
\alpha\in B_{n(p)} = \mathbb{Z}[U]/F_p(U)\\
\zeta\in A_p = \mathbb{Z}[T]/\Phi_p(T)
\end{array}
\quad\Big|\ \text{normaux} \qquad \Big|\ p \text{ premier impair}
\]\[(\zeta-1)^{p-1} \equiv 0 \quad (p) \quad\text{dans } A_{(p)} = A_p\]
LaTeX source
\[
(\zeta-1)^{p-1} \equiv 0 \quad (p) \quad\text{dans } A_{(p)} = A_p
\]\[\begin{aligned}
(\zeta-1)^{p-1} &= (\zeta-1)^{p-1} - F_p(\zeta)\\
&= p\,\zeta^{p-2} + \Bigl(\tfrac{(p-1)(p-2)}{2}-1\Bigr)\zeta^{p-3}\\
&\qquad + \Bigl(-\tfrac{(p-1)(p-2)(p-3)}{6}-1\Bigr)\zeta^{p-4} - \cdots
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
(\zeta-1)^{p-1} &= (\zeta-1)^{p-1} - F_p(\zeta)\\
&= p\,\zeta^{p-2} + \Bigl(\tfrac{(p-1)(p-2)}{2}-1\Bigr)\zeta^{p-3}\\
&\qquad + \Bigl(-\tfrac{(p-1)(p-2)(p-3)}{6}-1\Bigr)\zeta^{p-4} - \cdots
\end{aligned}
\]\[\frac1p(\zeta-1)^{p-1} = \zeta^{p-2} + \cdots +\]
LaTeX source
\[
\frac1p(\zeta-1)^{p-1} = \zeta^{p-2} + \cdots +
\]\[N\Bigl(\frac1p(\zeta-1)^{p-1}\Bigr) = \frac{1}{p^{p-1}}\, N(\zeta-1)^{p-1}.\]
LaTeX source
\[
N\Bigl(\frac1p(\zeta-1)^{p-1}\Bigr) = \frac{1}{p^{p-1}}\, N(\zeta-1)^{p-1}.
\]\[N(\zeta-1) = \prod_1^{p-1}(\zeta_i-1) = \text{\struck{$\varepsilon$}}\prod(1-\zeta_i) = F_p(1) = \underbrace{1+\cdots+1}_{} = p\]
LaTeX source
\[
N(\zeta-1) = \prod_1^{p-1}(\zeta_i-1) = \text{\struck{$\varepsilon$}}\prod(1-\zeta_i) = F_p(1) = \underbrace{1+\cdots+1}_{} = p
\]\[N\Bigl(\frac1p(\zeta-1)^{p-1}\Bigr) = 1\]
LaTeX source
\[
N\Bigl(\frac1p(\zeta-1)^{p-1}\Bigr) = 1
\]\[\boxed{(\zeta-1)^{p-1} = p\,u} \qquad u \text{ \emph{unité}}\]
LaTeX source
\[
\boxed{(\zeta-1)^{p-1} = p\,u} \qquad u \text{ \emph{unité}}
\]\[A/(\zeta-1)A = (\underbrace{A/pA}_{A_0})/(\zeta-1)A_0 = \bigl(\mathbb{F}_p[T]/(T-1)^{p-1}\bigr)/(T-1) = \mathbb{F}_p\]
LaTeX source
\[
A/(\zeta-1)A = (\underbrace{A/pA}_{A_0})/(\zeta-1)A_0 = \bigl(\mathbb{F}_p[T]/(T-1)^{p-1}\bigr)/(T-1) = \mathbb{F}_p
\]\[(\alpha-2)^{\varphi'} = p\,u\]
LaTeX source
\[
(\alpha-2)^{\varphi'} = p\,u
\]\[N(\alpha-2)^{\varphi'} = p^{\varphi'} N(u)\]
LaTeX source
\[
N(\alpha-2)^{\varphi'} = p^{\varphi'} N(u)
\]\[N(\alpha-2) = \prod(\alpha_i-2) = (-1)^{\varphi'}\prod(2-\alpha_i) = (-1)^{\varphi'} \underbrace{F_{p^r}(2)}_{p} = (-1)^{\varphi'}\,\Phi_{p^r}(1)\]
LaTeX source
\[
N(\alpha-2) = \prod(\alpha_i-2) = (-1)^{\varphi'}\prod(2-\alpha_i) = (-1)^{\varphi'} \underbrace{F_{p^r}(2)}_{p} = (-1)^{\varphi'}\,\Phi_{p^r}(1)
\]\[\mathbb{F}_p[T]/\Phi_n(T) = \mathbb{F}_p[T]/\text{\struck{$(T-1)^{\varphi(p^r)}$}}\ \Phi_{n'}(T)^{\varphi(p^r)}\]
LaTeX source
\[
\mathbb{F}_p[T]/\Phi_n(T) = \mathbb{F}_p[T]/\text{\struck{$(T-1)^{\varphi(p^r)}$}}\ \Phi_{n'}(T)^{\varphi(p^r)}
\]\[\text{\struck{$=\mathbb{F}_p[T]/\Phi_{n'}(T^{\cdots})$}} \quad \prod_{(r,n')=1}(T-\xi_1^r)\Big/\prod(T-\zeta_i)^{\varphi(p^n)}\]
LaTeX source
\[
\text{\struck{$=\mathbb{F}_p[T]/\Phi_{n'}(T^{\cdots})$}} \quad \prod_{(r,n')=1}(T-\xi_1^r)\Big/\prod(T-\zeta_i)^{\varphi(p^n)}
\]\[N_{A_n/A_{n'}}(\zeta-1)\]
LaTeX source
\[
N_{A_n/A_{n'}}(\zeta-1)
\]\[\underbrace{\mathbb{F}_p[T']/\Phi_{n'}(T')}\,[T]/(T^{p^n}-T') \xrightarrow{\ \text{cf.}\ } A_n\otimes\mathbb{F}_p\]
LaTeX source
\[
\underbrace{\mathbb{F}_p[T']/\Phi_{n'}(T')}\,[T]/(T^{p^n}-T') \xrightarrow{\ \text{cf.}\ } A_n\otimes\mathbb{F}_p
\]\[(\alpha-2)^{p'} \equiv 0 \quad (p)\]
LaTeX source
\[
(\alpha-2)^{p'} \equiv 0 \quad (p)
\]\[(\alpha-2)^{p'} = p\,u\]
LaTeX source
\[
(\alpha-2)^{p'} = p\,u
\]\[N(\alpha-2)^{p'} = p^{p'} N(u)\]
LaTeX source
\[
N(\alpha-2)^{p'} = p^{p'} N(u)
\]\[N(\alpha-2) = \prod(\alpha_i-2) = (-1)^{p'}\prod(2-\alpha_i) = (-1)^{p'} F_p(2) = (-1)^{p'}\Phi_p(1) = (-1)^{p'}p\]
LaTeX source
\[
N(\alpha-2) = \prod(\alpha_i-2) = (-1)^{p'}\prod(2-\alpha_i) = (-1)^{p'} F_p(2) = (-1)^{p'}\Phi_p(1) = (-1)^{p'}p
\]\[(\zeta-1)(\zeta^{-1}-1) = 1+1-\alpha\]
LaTeX source
\[
(\zeta-1)(\zeta^{-1}-1) = 1+1-\alpha
\]\[B[T]/(T^2-\alpha T+1) \xrightarrow{\ \text{iso ?}\ } A\]
LaTeX source
\[
B[T]/(T^2-\alpha T+1) \xrightarrow{\ \text{iso ?}\ } A
\]\[(\zeta-1)^{\varphi} \equiv 0 \quad (p)\]
LaTeX source
\[
(\zeta-1)^{\varphi} \equiv 0 \quad (p)
\]\[(\xi-1)^{\varphi} = p\,u\]
LaTeX source
\[
(\xi-1)^{\varphi} = p\,u
\]\[N(\xi-1)^{\varphi} = p^{\varphi} N(u)\]
LaTeX source
\[
N(\xi-1)^{\varphi} = p^{\varphi} N(u)
\]\[\prod(\zeta_i-1) = (-1)^{\varphi}\prod_i(1-\zeta_i) = (-1)^{\varphi}\,\underbrace{\Phi_{p^r}(1)}_{p}\]
LaTeX source
\[
\prod(\zeta_i-1) = (-1)^{\varphi}\prod_i(1-\zeta_i) = (-1)^{\varphi}\,\underbrace{\Phi_{p^r}(1)}_{p}
\]\[\begin{cases}
\displaystyle\prod_{\substack{d\mid p^r\\ d\neq 1}} \Phi_d(T) = \text{\struck{$T^n$}}\ 1+T+\cdots+T^{p^r-1} = p^r \quad p\\
\text{\struck{$\displaystyle\prod_{d\mid p^r} p = \Phi_p(1)\Phi_{p^2}(1)\cdots$}} \quad p\,p^?-p^r\\
\Phi_{p^r}(1) = p
\end{cases}\]
LaTeX source
\[
\begin{cases}
\displaystyle\prod_{\substack{d\mid p^r\\ d\neq 1}} \Phi_d(T) = \text{\struck{$T^n$}}\ 1+T+\cdots+T^{p^r-1} = p^r \quad p\\
\text{\struck{$\displaystyle\prod_{d\mid p^r} p = \Phi_p(1)\Phi_{p^2}(1)\cdots$}} \quad p\,p^?-p^r\\
\Phi_{p^r}(1) = p
\end{cases}
\]\[\xi\in A_{n'n''=n} \supset A_{n''}\ni\zeta'' \qquad ((n',n'')=1)\]
LaTeX source
\[
\xi\in A_{n'n''=n} \supset A_{n''}\ni\zeta'' \qquad ((n',n'')=1)
\]\[\cup \qquad \zeta'\in A_{n'}\]
LaTeX source
\[
\cup \qquad \zeta'\in A_{n'}
\]\[G = (\mathbb{Z}/n\mathbb{Z})^* \simeq G'\times G'', \qquad G'\simeq(\mathbb{Z}/n'\mathbb{Z})^*,\quad G''\simeq(\mathbb{Z}/n''\mathbb{Z})^*\]
LaTeX source
\[
G = (\mathbb{Z}/n\mathbb{Z})^* \simeq G'\times G'', \qquad G'\simeq(\mathbb{Z}/n'\mathbb{Z})^*,\quad G''\simeq(\mathbb{Z}/n''\mathbb{Z})^*
\]\[\underbrace{A_{n'}\otimes_{\mathbb{Z}} A_{n''}} \xrightarrow{\ \sim\ } A_{n'n''}\]
LaTeX source
\[
\underbrace{A_{n'}\otimes_{\mathbb{Z}} A_{n''}} \xrightarrow{\ \sim\ } A_{n'n''}
\]\[\zeta_i\eta_j + \zeta_i^{-1}\eta_j^{-1}\]
LaTeX source
\[
\zeta_i\eta_j + \zeta_i^{-1}\eta_j^{-1}
\]\[B_n[T]/(T^2-\alpha T+1) \longrightarrow A_n \qquad
\begin{cases} \alpha_n\longmapsto \xi_n+\xi_n^{-1}\\ T\longmapsto \xi_n\end{cases}\]
LaTeX source
\[
B_n[T]/(T^2-\alpha T+1) \longrightarrow A_n \qquad
\begin{cases} \alpha_n\longmapsto \xi_n+\xi_n^{-1}\\ T\longmapsto \xi_n\end{cases}
\]\[\Big|\quad
\begin{aligned}
&\mu_n\supset\mu_n^* = \operatorname{Spec}\mathbb{Z}[T]/\Phi_n(T)\\
&\mu_{nn'}^*\simeq\mu_n^*\times\mu_{n'}^*\\
&\mu_n^*/\{\pm1\} = \tilde\mu_n^*
\end{aligned}\]
LaTeX source
\[
\Big|\quad
\begin{aligned}
&\mu_n\supset\mu_n^* = \operatorname{Spec}\mathbb{Z}[T]/\Phi_n(T)\\
&\mu_{nn'}^*\simeq\mu_n^*\times\mu_{n'}^*\\
&\mu_n^*/\{\pm1\} = \tilde\mu_n^*
\end{aligned}
\]\[B_n = A_n^{\sigma}\]
LaTeX source
\[
B_n = A_n^{\sigma}
\]\[\begin{aligned}
\sigma(F+GT) &= F+G(U-T)\\
&= (F+GU)-GT
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\sigma(F+GT) &= F+G(U-T)\\
&= (F+GU)-GT
\end{aligned}
\]\[\sigma(F+GT) = F+GT \iff
\begin{cases} GU=0 & G=-G\\ \iff G=0 \end{cases}\]
LaTeX source
\[
\sigma(F+GT) = F+GT \iff
\begin{cases} GU=0 & G=-G\\ \iff G=0 \end{cases}
\]\[\begin{array}{l}
U = T+T^{-1}\\
\sigma(T) = T^{-1}
\end{array}
\ \Big|\
F_n(U) = F_n(T+T^{-1}) = T^{-\varphi'}\Phi_n(T)\]
LaTeX source
\[
\begin{array}{l}
U = T+T^{-1}\\
\sigma(T) = T^{-1}
\end{array}
\ \Big|\
F_n(U) = F_n(T+T^{-1}) = T^{-\varphi'}\Phi_n(T)
\]\[A \xleftarrow{\ \sim\ } B[S]/(S^2-US+1) \qquad\text{\struck{$\mathbb{Z}[T]/(T^n-1)$}}\]
LaTeX source
\[
A \xleftarrow{\ \sim\ } B[S]/(S^2-US+1) \qquad\text{\struck{$\mathbb{Z}[T]/(T^n-1)$}}
\]\[= B[\dot S] = B.1\oplus B.\dot S\]
LaTeX source
\[ = B[\dot S] = B.1\oplus B.\dot S \]
\[\text{\struck{$F+T$}}\quad F(U)+T\,G(U) = 0 \ \Longrightarrow\ F(U)+T^{-1}G(U) = 0\]
LaTeX source
\[
\text{\struck{$F+T$}}\quad F(U)+T\,G(U) = 0 \ \Longrightarrow\ F(U)+T^{-1}G(U) = 0
\]\[T\,G(U) = T^{-1}G(U), \qquad (T-T^{-1})\,G(U) = 0 \qquad G(U) = 0,\]
LaTeX source
\[
T\,G(U) = T^{-1}G(U), \qquad (T-T^{-1})\,G(U) = 0 \qquad G(U) = 0,
\]