Exercise 4 (Candidates who have followed the specialization course)
Two column matrices $\binom { x } { y }$ and $\binom { x ^ { \prime } } { y ^ { \prime } }$ with integer coefficients are said to be congruent modulo 5 if and only if $\left\{ \begin{array} { l } x \equiv x ^ { \prime } [ 5 ] \\ y \equiv y ^ { \prime } [ 5 ] \end{array} \right.$. Two square matrices of order $2 \left( \begin{array} { l l } a & c \\ b & d \end{array} \right)$ and $\left( \begin{array} { l l } a ^ { \prime } & c ^ { \prime } \\ b ^ { \prime } & d ^ { \prime } \end{array} \right)$ with integer coefficients are said to be congruent modulo 5 if and only if $\left\{ \begin{array} { l } a \equiv a ^ { \prime } [ 5 ] \\ b \equiv b ^ { \prime } [ 5 ] \\ c \equiv c ^ { \prime } [ 5 ] \\ d \equiv d ^ { \prime } [ 5 ] \end{array} \right.$.
Alice and Bob want to exchange messages using the procedure described below.

• They choose a square matrix M of order 2, with integer coefficients.
• Their initial message is written in capital letters without accents.
• Each letter of this message is replaced by a column matrix $\binom { x } { y }$ deduced from the table below: $x$ is the digit located at the top of the column and $y$ is the digit located to the left of the row; for example, the letter T in an initial message corresponds to the column matrix $\binom { 4 } { 3 }$.
• We calculate a new matrix $\binom { x ^ { \prime } } { y ^ { \prime } }$ by multiplying $\binom { x } { y }$ on the left by the matrix $M$:

$$\binom { x ^ { \prime } } { y ^ { \prime } } = \mathrm { M } \binom { x } { y } .$$

	0	1	2	3	4
0	A	B	C	D	E
1	F	G	H	I	J
2	K	L	M	N	O
3	P	Q	R	S	T
4	U	V	X	Y	Z

There is a field with 10 elements.

There is a field with 121 elements.

Every group of order 6 abelian.

Pick the correct statement(s) below.
(a) There exists a group of order 44 with a subgroup isomorphic to $\mathbb { Z } / 2 \oplus \mathbb { Z } / 2$.
(b) There exists a group of order 44 with a subgroup isomorphic to $\mathbb { Z } / 4$.
(c) There exists a group of order 44 with a subgroup isomorphic to $\mathbb { Z } / 2 \oplus \mathbb { Z } / 2$ and a subgroup isomorphic to $\mathbb { Z } / 4$.
(d) There exists a group of order 44 without any subgroup isomorphic to $\mathbb { Z } / 2 \oplus \mathbb { Z } / 2$ or to $\mathbb { Z } / 4$.

Let $F$ be a field with 256 elements, and $f \in F [ x ]$ a polynomial with all its roots in $F$. Then,
(a) $f \neq x ^ { 15 } - 1$;
(b) $f \neq x ^ { 63 } - 1$;
(c) $f \neq x ^ { 2 } + x + 1$;
(d) if $f$ has no multiple roots, then $f$ is a factor of $x ^ { 256 } - x$.

Let $m$ and $n$ be positive integers and $p$ a prime number. Let $G \subseteq \mathrm{GL}_{m}(\mathbb{F}_{p})$ be a subgroup of order $p^{n}$. Let $U \subseteq \mathrm{GL}_{m}(\mathbb{F}_{p})$ be the subgroup that consists of all the matrices with 1's on the diagonal and 0's below the diagonal. Show that there exists $A \in \mathrm{GL}_{m}(\mathbb{F}_{p})$ such that $AGA^{-1} \subseteq U$.

Let $G$ be a subgroup of the group of permutations on a finite set $X$. Let $F$ be the $\mathbb{C}$-vector-space of all the functions from $X$ to $\mathbb{C}$. $G$ acts on $F$ by $(g \cdot f) : x \mapsto f(g^{-1}(x))$. Show that there is a $\phi \in F$ such that $g \cdot \phi = \phi$ for every $g \in G$. Show that there is a subspace $F'$ of $F$ such that $F = F' \oplus \mathbb{C}\langle \phi \rangle$ and such that $g \cdot f \in F'$ for every $g \in G$ and $f \in F'$.

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb{k})$ where $\mathbb{k}$ is an algebraically closed field. Choose the correct statement(s) from below:
(A) Every element of $G$ is diagonalizable;
(B) Every element of $G$ is diagonalizable if $\mathbb{k}$ is an algebraic closure of $\mathbb{Q}$;
(C) Every element of $G$ is diagonalizable if $\mathbb{k}$ is an algebraic closure of $\mathbb{F}_p$;
(D) There exists a basis of $\mathbb{k}^n$ with respect to which every element of $G$ is a diagonal matrix.

Which of the following can not be the class equation for a group of appropriate order?
(A) $14 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 7$.
(B) $18 = 1 + 1 + 1 + 1 + 2 + 3 + 9$.
(C) $6 = 1 + 2 + 3$.
(D) $31 = 1 + 3 + 6 + 6 + 7 + 8$.

Let $U ( n )$ be the group of $n \times n$ unitary complex matrices. Let $P \subset U ( n )$ be the set of all finite order elements of $U ( n )$, that is, $P = \left\{ X \in U ( n ) \mid X ^ { m } = 1 \text{ for some } m \geq 1 \right\}$. Show that $P$ is dense in $U ( n )$.

Let $A$ be a non-trivial subgroup of $\mathbb { R }$ generated by finitely many elements. Let $r$ be a real number such that $x \longrightarrow r x$ is an automorphism of $A$. Show that $r$ and $r ^ { - 1 }$ are zeros of monic polynomials with integer coefficients.

Show that $d_{n} \geqslant 1$.

We assume in this question that the space $E$ has dimension 1. Show that the root systems of $E$ are the sets $\{ \alpha , - \alpha \}$, with $\alpha \in E \backslash \{ 0 \}$.

In this question, the space $E$ has dimension $n \geq 2$. For every pair $(\alpha , \beta)$ of non-zero vectors of $E$, let $\theta _ { \alpha , \beta }$ be the geometric angle between $\alpha$ and $\beta$, that is, the unique element of $[ 0 , \pi ]$ given by: $\| \alpha \| . \| \beta \| \cos \theta _ { \alpha , \beta } = \langle \alpha , \beta \rangle$.
Let $\mathcal { R }$ be a root system of $E$ and let $\alpha , \beta$ be two non-collinear elements of $\mathcal { R }$.
a) Show, using property 4, that: $2 \frac { \| \alpha \| } { \| \beta \| } \left| \cos \theta _ { \alpha , \beta } \right| .2 \frac { \| \beta \| } { \| \alpha \| } \left| \cos \theta _ { \alpha , \beta } \right| \leq 3$.
b) Assume $\| \alpha \| \leq \| \beta \|$. Show that the pair $(\alpha , \beta)$ is found in one of the configurations listed in the table below (each row corresponding to a configuration):

$\theta _ { \alpha , \beta }$	$\cos \theta _ { \alpha , \beta }$	$\| \beta \| / \| \alpha \|$
$\pi / 2$	0	$\geq 1$
$\pi / 3$	$1 / 2$	1
$2 \pi / 3$	$- 1 / 2$	1
$\pi / 4$	$\sqrt { 2 } / 2$	$\sqrt { 2 }$
$3 \pi / 4$	$- \sqrt { 2 } / 2$	$\sqrt { 2 }$
$\pi / 6$	$\sqrt { 3 } / 2$	$\sqrt { 3 }$
$5 \pi / 6$	$- \sqrt { 3 } / 2$	$\sqrt { 3 }$

In this question, the space $E$ has dimension $n \geq 2$. Conversely, assume that a pair $(\alpha , \beta)$ of non-collinear vectors of $E$ is found in one of the configurations listed in the table below. Show that the real number $2 \frac { \langle \alpha , \beta \rangle } { \langle \alpha , \alpha \rangle }$ is an integer; specify its value.

$\theta _ { \alpha , \beta }$	$\cos \theta _ { \alpha , \beta }$	$\| \beta \| / \| \alpha \|$
$\pi / 2$	0	$\geq 1$
$\pi / 3$	$1 / 2$	1
$2 \pi / 3$	$- 1 / 2$	1
$\pi / 4$	$\sqrt { 2 } / 2$	$\sqrt { 2 }$
$3 \pi / 4$	$- \sqrt { 2 } / 2$	$\sqrt { 2 }$
$\pi / 6$	$\sqrt { 3 } / 2$	$\sqrt { 3 }$
$5 \pi / 6$	$- \sqrt { 3 } / 2$	$\sqrt { 3 }$

In this question, the space $E$ has dimension $n = 2$. For every root system $\mathcal { R }$ of $E$, we set $$\theta _ { \mathcal { R } } = \min \left\{ \theta _ { \alpha , \beta } \mid ( \alpha , \beta ) \in \mathcal { R } ^ { 2 } , \alpha \neq \beta \text { and } \alpha \neq - \beta \right\}$$ Show that $\theta _ { \mathcal { R } }$ is well-defined and equals $\pi / 2 , \pi / 3 , \pi / 4$ or $\pi / 6$.

In this question, the space $E$ has dimension $n = 2$. For every root system $\mathcal { R }$ of $E$, we set $$\theta _ { \mathcal { R } } = \min \left\{ \theta _ { \alpha , \beta } \mid ( \alpha , \beta ) \in \mathcal { R } ^ { 2 } , \alpha \neq \beta \text { and } \alpha \neq - \beta \right\}$$ For each value of $k \in \{ 2,3,4,6 \}$, draw graphically a root system $\mathcal { R } _ { k }$ such that $\theta _ { \mathcal { R } _ { k } } = \pi / k$. It is not necessary to justify that the figures drawn represent root systems. What is the cardinality of $\mathcal { R } _ { k }$? No justification is required.

In this question, the space $E$ has dimension $n = 3$. Let $(e _ { 1 } , e _ { 2 } , e _ { 3 })$ be an orthonormal basis of $E$ and $\mathcal { R } _ { 0 } = \left\{ e _ { i } - e _ { j } \mid 1 \leq i , j \leq 3 , i \neq j \right\}$.
Show that the vector subspace of $E$ spanned by the set $\mathcal { R } _ { 0 }$ is a vector plane.

In this question, the space $E$ has dimension $n = 3$. Let $(e _ { 1 } , e _ { 2 } , e _ { 3 })$ be an orthonormal basis of $E$ and $\mathcal { R } _ { 0 } = \left\{ e _ { i } - e _ { j } \mid 1 \leq i , j \leq 3 , i \neq j \right\}$.
Draw graphically $\mathcal { R } _ { 0 }$ in the plane $\operatorname { Vect } \left( \mathcal { R } _ { 0 } \right)$. Recognize one of the root systems represented in question I.D.2.

We equip $\mathbb{C}[X]$ with the internal composition law given by composition, denoted $\circ$. We denote by $G$ the set of complex polynomials of degree 1.
Verify that $G$ is a group for the law $\circ$.

For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $\mathcal{C}(P)$ the set of complex polynomials that commute with the polynomial $P$ under composition. Every non-constant polynomial commuting with $P_\alpha$ is monic.
Deduce that, for every integer $n \geqslant 1$, there exists at most one polynomial of degree $n$ that commutes with $P_\alpha$. Determine $\mathcal{C}(X^2)$.

For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $G$ the set of complex polynomials of degree 1, and the inverse of $U \in G$ under composition is denoted $U^{-1}$.
Let $P$ be a complex polynomial of degree 2. Justify the existence and uniqueness of $U \in G$ and $\alpha \in \mathbb{C}$ such that $U \circ P \circ U^{-1} = P_\alpha$. Determine these two elements when $P = T_2$.

For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $\mathcal{C}(P)$ the set of complex polynomials that commute with $P$ under composition. The Chebyshev polynomials $(T_n)_{n \in \mathbb{N}}$ are defined by $T_n(\cos\theta) = \cos(n\theta)$.
Justify that $\mathcal{C}(T_2) = \{-1/2\} \cup \{T_n, n \in \mathbb{N}\}$.

For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $\mathcal{C}(P_\alpha)$ the set of complex polynomials that commute with $P_\alpha$ under composition.
Show that the only complex numbers $\alpha$ such that $\mathcal{C}(P_\alpha)$ contains a polynomial of degree three are 0 and $-2$.