a) Find a polynomial $p ( x )$ with real coefficients such that $p ( \sqrt { 2 } + i ) = 0$. b) Find a polynomial $q ( x )$ with rational coefficients and having the smallest possible degree such that $q ( \sqrt { 2 } + i ) = 0$. Show that any other polynomial with rational coefficients and having $\sqrt { 2 } + i$ as a root has $q ( x )$ as a factor.

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 $\mathcal{C}(P)$ 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 $\mathcal{C}(T_2) = \{-1/2\} \cup \{T_n, n \in \mathbb{N}\}$.

Show that: $X^p + \alpha_{p-1}X^{p-1} + \cdots + \alpha_0$ divides the polynomial $\chi_f$.

Prove that $\chi_f(f)$ is the zero endomorphism.

Show that: $X^p + \alpha_{p-1}X^{p-1} + \cdots + \alpha_0$ divides the polynomial $\chi_f$.

Prove that $\chi_f(f)$ is the zero endomorphism.

(a) Let $A , B \in \mathbb { Q } [ X ]$ be two polynomials that have a common root in $\mathbb { C }$. Show that $A$ and $B$ are not coprime in $\mathbb { Q } [ X ]$.
(b) Show that the roots of $\Pi _ { \alpha }$ in $\mathbb { C }$ are simple.

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 $\lim _ { n \rightarrow + \infty } \alpha _ { n } = + \infty$.

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Let $$m = \min\{k \in \mathbb{N} \mid \exists P \in \mathscr{V}(A) \text{ with } \deg(P) = k\}$$ Show that there exists a unique polynomial $p \in \mathbb{C}[X]$ satisfying the three conditions
(i) $p \in \mathscr{V}(A)$,
(ii) $\deg(p) = m$,
(iii) $p$ is monic.

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. We assume throughout this part that $A$ is diagonalizable in $\mathscr{M}_n(\mathbb{C})$ and we denote by $\lambda_1, \cdots, \lambda_\ell$ its eigenvalues with $\lambda_i \neq \lambda_j$ if $i \neq j$. Show that $$\varphi_A(X) = (X - \lambda_1) \cdots (X - \lambda_\ell).$$

Show that $p \wedge p_0 = 1$ if and only if $p$ has no stable root.