Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
Show that a matrix is nilpotent if, and only if, its characteristic polynomial is equal to $X^n$.

Let $z \in \mathbb { P } _ { n }$ and let $p$ be a prime number not dividing $n$.
(a) Express as a function of $n$ the number $\prod _ { 1 \leq i < j \leq n } \left( z _ { i } - z _ { j } \right) ^ { 2 }$, where $z _ { 1 } , z _ { 2 } , \ldots , z _ { n }$ are the roots of the polynomial $P = X ^ { n } - 1$. Hint: One may consider the numbers $P ^ { \prime } \left( z _ { i } \right)$.
(b) Show that $\Pi _ { z } \left( z ^ { p } \right) = 0$. Hint: show that if $\Pi _ { z } \left( z ^ { p } \right) \neq 0$, then there exists an algebraic integer $u$ such that $n ^ { n } = u \cdot \Pi _ { z } \left( z ^ { p } \right)$.
(c) Conclude that $\Phi _ { n } = \Pi _ { z }$.

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
Show the converse of question 12: if 0 is the unique eigenvalue of $A$, then $A$ is nilpotent.

A monic polynomial of degree $d \geq 1$ $$P = \sum _ { i = 0 } ^ { d } a _ { i } X ^ { i } \in \mathbb { C } [ X ]$$ is called reciprocal if $a _ { i } = a _ { d - i }$ for $0 \leq i \leq d$.
(a) Show that a monic polynomial $P \in \mathbb { C } [ X ]$ of degree $d$ is reciprocal if and only if $X ^ { d } P \left( \frac { 1 } { X } \right) = P$.
(b) Let $P \in \mathbb { C } [ X ]$ be a monic reciprocal polynomial. Show that if $x \in \mathbb { C }$ is a root of $P$, then $x \neq 0$ and $\frac { 1 } { x }$ is also a root of $P$, with the same multiplicity.

If $\alpha$ is an algebraic number with minimal polynomial $\Pi _ { \alpha }$, the complex roots of $\Pi _ { \alpha }$ different from $\alpha$ are called the conjugates of $\alpha$. We denote by $C ( \alpha )$ the set of conjugates of $\alpha$.
Let $x$ be an algebraic number of modulus 1 and such that $x \notin \{ - 1,1 \}$. Show that $\frac { 1 } { x }$ is a conjugate of $x$. Deduce that $\Pi _ { x }$ is reciprocal.

In this part, we assume $n \geqslant 2$. Let $J \in \mathcal{M}_{n}(\mathbb{R})$ be the matrix canonically associated with the endomorphism $\varphi \in \mathcal{L}(\mathbb{R}^{n})$ defined by $\varphi: e_{j} \mapsto e_{j+1}$ if $j \in \{1, \ldots, n-1\}$ and $\varphi(e_{n}) = e_{1}$, where $(e_{1}, \ldots, e_{n})$ is the canonical basis of $\mathbb{R}^{n}$.
Determine the characteristic polynomial of $J$.

We denote by $\mathcal { S }$ the set of real numbers $\alpha \in ] 1 , + \infty [$ which are also algebraic integers of degree at least 2 and which satisfy $\max _ { \gamma \in C ( \alpha ) } | \gamma | = 1$.
Let $\alpha$ be an element of $\mathcal { S }$ and let $\gamma \in C ( \alpha )$ have modulus 1.
(a) Show that the minimal polynomial of $\alpha$ is reciprocal and that $\frac { 1 } { \alpha }$ is a conjugate of $\alpha$.
(b) Show that $\gamma$ is not a root of unity.
(c) Show that all conjugates of $\alpha$ other than $\frac { 1 } { \alpha }$ have modulus 1.

We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free. There exists a basis $\mathcal{B} = (u_1, \ldots, u_n)$ of $E$ in which $f$ has a block diagonal matrix with Jordan blocks of sizes $m_k$ associated to eigenvalues $\lambda_k$. We set $x_0 = u_1 + u_{m_1+1} + \cdots + u_{m_1 + \cdots + m_{p-1}+1}$.
Determine the polynomials $Q \in \mathbb{C}[X]$ such that $Q(f)(x_0) = 0$.

We denote by $\mathcal { S }$ the set of real numbers $\alpha \in ] 1 , + \infty [$ which are also algebraic integers of degree at least 2 and which satisfy $\max _ { \gamma \in C ( \alpha ) } | \gamma | = 1$.
Show that the degree of every element of $\mathcal { S }$ is an even integer, greater than or equal to 4.

We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free. There exists a basis $\mathcal{B} = (u_1, \ldots, u_n)$ of $E$ in which $f$ has a block diagonal matrix with Jordan blocks of sizes $m_k$ associated to eigenvalues $\lambda_k$. We set $x_0 = u_1 + u_{m_1+1} + \cdots + u_{m_1 + \cdots + m_{p-1}+1}$.
Justify that $f$ is cyclic.

In this part, we assume $n \geqslant 2$. Let $J \in \mathcal{M}_{n}(\mathbb{R})$ be the matrix canonically associated with the endomorphism $\varphi \in \mathcal{L}(\mathbb{R}^{n})$ defined by $\varphi: e_{j} \mapsto e_{j+1}$ if $j \in \{1, \ldots, n-1\}$ and $\varphi(e_{n}) = e_{1}$, where $(e_{1}, \ldots, e_{n})$ is the canonical basis of $\mathbb{R}^{n}$.
Determine the complex eigenvalues of $J$ and the associated eigenspaces.

For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ Show that if $x \in \mathbb { C }$ is a root of $P _ { n }$, then $\frac { 1 } { x }$ is also a root of $P _ { n }$, with the same multiplicity.

We assume that $f$ is cyclic and we choose a vector $x_0$ in $E$ such that $(x_0, f(x_0), \ldots, f^{n-1}(x_0))$ is a basis of $E$. Let $g \in \mathcal{C}(f)$, an endomorphism that commutes with $f$.
Justify the existence of $\lambda_0, \lambda_1, \ldots, \lambda_{n-1}$ of $\mathbb{K}$ such that
$$g(x_0) = \sum_{k=0}^{n-1} \lambda_k f^k(x_0)$$

For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ We denote by $\alpha _ { n } , \frac { 1 } { \alpha _ { n } } , \gamma _ { n } , \frac { 1 } { \gamma _ { n } }$ the roots of $P _ { n }$ in $\mathbb { C }$ and we set $$t _ { n } = \alpha _ { n } + \frac { 1 } { \alpha _ { n } } , \quad s _ { n } = \gamma _ { n } + \frac { 1 } { \gamma _ { n } } .$$ Show that $t _ { n } + s _ { n } = 6 + n$ and $t _ { n } s _ { n } = 8 + n$.

We assume that $f$ is cyclic and we choose a vector $x_0$ in $E$ such that $(x_0, f(x_0), \ldots, f^{n-1}(x_0))$ is a basis of $E$. Let $g \in \mathcal{C}(f)$, an endomorphism that commutes with $f$, and suppose $g(x_0) = \sum_{k=0}^{n-1} \lambda_k f^k(x_0)$.
Show then that $g \in \mathbb{K}[f]$.

For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ We denote by $\alpha _ { n } , \frac { 1 } { \alpha _ { n } } , \gamma _ { n } , \frac { 1 } { \gamma _ { n } }$ the roots of $P _ { n }$ in $\mathbb { C }$ and we set $$t _ { n } = \alpha _ { n } + \frac { 1 } { \alpha _ { n } } , \quad s _ { n } = \gamma _ { n } + \frac { 1 } { \gamma _ { n } } .$$ Show that $s _ { n }$ is real and that $0 < s _ { n } < 2$. Deduce that $\gamma _ { n }$ is not real and that $\gamma _ { n }$ has modulus 1.

We assume that $f$ is cyclic. Establish that $g \in \mathcal{C}(f)$ if and only if there exists a polynomial $R \in \mathbb{K}_{n-1}[X]$ such that $g = R(f)$.

For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ We denote by $\alpha _ { n } , \frac { 1 } { \alpha _ { n } } , \gamma _ { n } , \frac { 1 } { \gamma _ { n } }$ the roots of $P _ { n }$ in $\mathbb { C }$ and we set $$t _ { n } = \alpha _ { n } + \frac { 1 } { \alpha _ { n } } , \quad s _ { n } = \gamma _ { n } + \frac { 1 } { \gamma _ { n } } .$$ We denote by $\mathcal { S }$ the set of real numbers $\alpha \in ] 1 , + \infty [$ which are also algebraic integers of degree at least 2 and which satisfy $\max _ { \gamma \in C ( \alpha ) } | \gamma | = 1$.
(a) Show that $t _ { n }$ and $s _ { n }$ are irrational.
(b) Deduce that $P _ { n }$ is irreducible in $\mathbb { Q } [ X ]$ and that $\alpha _ { n } \in \mathcal { S }$.
(c) Show that $\lim _ { n \rightarrow + \infty } \alpha _ { n } = + \infty$.

For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ We denote by $\mathcal { S }$ the set of real numbers $\alpha \in ] 1 , + \infty [$ which are also algebraic integers of degree at least 2 and which satisfy $\max _ { \gamma \in C ( \alpha ) } | \gamma | = 1$. Let $\mathcal { T }$ be the set of $\alpha \in \mathcal { S }$ of degree 4. Show that $\mathcal { T }$ has a smallest element and calculate this number.

We denote by $d$ the degree of $\pi_f$. Justify the existence of a vector $x_1$ of $E$ such that $\left(x_1, f(x_1), \ldots, f^{d-1}(x_1)\right)$ is free.

We denote by $d$ the degree of $\pi_f$, and $x_1$ is a vector of $E$ such that $\left(x_1, f(x_1), \ldots, f^{d-1}(x_1)\right)$ is free. We set $e_1 = x_1, e_2 = f(x_1), \ldots, e_d = f^{d-1}(x_1)$ and $E_1 = \operatorname{Vect}(e_1, e_2, \ldots, e_d)$.
Show that $E_1$ is stable under $f$ and that $E_1 = \{P(f)(x_1) \mid P \in \mathbb{K}[X]\}$.

We denote by $d$ the degree of $\pi_f$, $E_1 = \operatorname{Vect}(e_1, e_2, \ldots, e_d)$ where $e_i = f^{i-1}(x_1)$. We complete $(e_1, e_2, \ldots, e_d)$ to a basis $(e_1, e_2, \ldots, e_n)$ of $E$. Let $\Phi$ be the $d$-th coordinate form which associates to any vector $x$ of $E$ its coordinate along $e_d$. We denote by $F = \{x \in E \mid \forall i \in \mathbb{N}, \Phi(f^i(x)) = 0\}$.
Show that $F$ is stable under $f$ and that $E_1$ and $F$ are in direct sum.

Let $P ( X )$ be a monic polynomial of degree $d \geqslant 1$ with complex coefficients which we write in the form: $$P ( X ) = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { d - 1 } X ^ { d - 1 } + X ^ { d }$$ We assume that $a _ { 0 } \neq 0$. We denote by $\lambda _ { 1 } , \ldots , \lambda _ { d } \in \mathbb { C }$ the roots of $P ( X )$ (with multiplicity). For all integers $n \geqslant 1$, we define: $$N _ { n } = \lambda _ { 1 } ^ { n } + \lambda _ { 2 } ^ { n } + \cdots + \lambda _ { d } ^ { n }$$
Let $Q ( X )$ be the reciprocal polynomial of $P ( X )$ defined by $Q ( X ) = X ^ { d } P \left( \frac { 1 } { X } \right)$. Show that: $$\begin{aligned} Q ( X ) & = 1 + a _ { d - 1 } X + \cdots + a _ { 1 } X ^ { d - 1 } + a _ { 0 } X ^ { d } \\ & = \left( 1 - \lambda _ { 1 } X \right) \left( 1 - \lambda _ { 2 } X \right) \cdots \left( 1 - \lambda _ { d } X \right) \end{aligned}$$

Let $P ( X )$ be a monic polynomial of degree $d \geqslant 1$ with complex coefficients which we write in the form: $$P ( X ) = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { d - 1 } X ^ { d - 1 } + X ^ { d }$$ We assume that $a _ { 0 } \neq 0$. We denote by $\lambda _ { 1 } , \ldots , \lambda _ { d } \in \mathbb { C }$ the roots of $P ( X )$ (with multiplicity). For all integers $n \geqslant 1$, we define: $$N _ { n } = \lambda _ { 1 } ^ { n } + \lambda _ { 2 } ^ { n } + \cdots + \lambda _ { d } ^ { n }$$
Let $Q ( X )$ be the reciprocal polynomial of $P ( X )$ defined by $Q ( X ) = X ^ { d } P \left( \frac { 1 } { X } \right)$. Show that: $$\begin{aligned} Q ( X ) & = 1 + a _ { d - 1 } X + \cdots + a _ { 1 } X ^ { d - 1 } + a _ { 0 } X ^ { d } \\ & = \left( 1 - \lambda _ { 1 } X \right) \left( 1 - \lambda _ { 2 } X \right) \cdots \left( 1 - \lambda _ { d } X \right) \end{aligned}$$

Let $P ( X )$ be a monic polynomial of degree $d \geqslant 1$ with complex coefficients which we write in the form: $$P ( X ) = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { d - 1 } X ^ { d - 1 } + X ^ { d }$$ We assume that $a _ { 0 } \neq 0$. We denote by $\lambda _ { 1 } , \ldots , \lambda _ { d } \in \mathbb { C }$ the roots of $P ( X )$ (with multiplicity). For all integers $n \geqslant 1$, we define: $$N _ { n } = \lambda _ { 1 } ^ { n } + \lambda _ { 2 } ^ { n } + \cdots + \lambda _ { d } ^ { n }$$
We define the function $f : \mathbb { R } \backslash \left( \mathbb { R } \cap \left\{ \frac { 1 } { \lambda _ { 1 } } , \ldots , \frac { 1 } { \lambda _ { d } } \right\} \right) \rightarrow \mathbb { C }$ by $f ( x ) = \frac { Q ^ { \prime } ( x ) } { Q ( x ) }$.
Show that there exists $r > 0$ such that $f$ is expandable as a power series on $] - r , r [$, and that the power series expansion of $f$ at 0 is written: $$\forall x \in ] - r , r \left[ , \quad f ( x ) = - \sum _ { n = 0 } ^ { \infty } N _ { n + 1 } x ^ { n } \right.$$