مسابقة دكتوراه 2026 — المدرسة الوطنية العليا للرياضيات

Exercice 1. (7 points)

Let $E$ be a normed space. We denote by $\mathcal{L}(E)$ the space of all bounded linear operators from $E$ into itself, i.e., $\mathcal{L}(E) = \{T : E \to E;\ T \text{ is linear and bounded}\}$. For $T \in \mathcal{L}(E)$ and $n \in \mathbb{N}$, we define $\ker(T^n) = \{x \in E;\ T^n(x) = 0\}$.

1. Let $T \in \mathcal{L}(E)$.

   (a) Show that $\ker(T^n)$ is closed for every $n \in \mathbb{N}$.

   (b) Suppose that $E$ is complete and that, for every $x \in E$, there exists $n \in \mathbb{N}$ such that $x \in \ker(T^n)$. Show that $T$ is nilpotent, i.e., there exists $d \in \mathbb{N}$ such that $T^d = 0$.

2. Let $E = \mathbb{C}[X]$ be the space of complex polynomials. For $P \in E$, define $\|P\| = \sum_{n=0}^{\infty} \|P^{(n)}\|_{\infty}$, where $\|P\|_{\infty} = \sup_{x \in [0,1]} |P(x)|$ and $P^{(n)}$ denotes the derivative of order $n$ of $P$.

   (a) Verify that $\|\cdot\|$ is a norm on $E$.

   (b) Let $T : E \longrightarrow E$, $\quad T(P) = P'$, for every $P \in E$. Verify that $T \in \mathcal{L}(E)$.

   (c) Show that for every $P \in E$, there exists $n \in \mathbb{N}$ such that $P \in \ker(T^n)$.

   (d) Show that $T$ is not nilpotent. Explain this result by comparing it with part 1. (b).

Exercice 2. (7 points)

Let $K = \mathbb{Q}(\sqrt{-23})$ and $\alpha = \dfrac{1 + \sqrt{-23}}{2}$.

1. Calculate the minimal polynomial of $\alpha$, the discriminant $\Delta_K$ of $K$ and the Minkowski constant $M_K$ of $K$.

2. Show that the ideals $\mathfrak{p} = (2, \alpha)$ and $\mathfrak{q} = (3, \alpha)$ are prime and not principal.

3. Give the factorization of $2\mathcal{O}_K$ and $3\mathcal{O}_K$ as a product of prime ideals.

4. Show that $\mathfrak{p}^3$ is principal.

5. Calculate $Cl(K)$.

6. Using the class group computed in question 5., show that there is no integers $x$ and $y$ such that $x^5 = 4y^2 + 23$.

Exercice 3. (6 points)

The goal of this exercise is to prove that every continuous map from the closed unit disk $\overline{D} = \{z \in \mathbb{C} :\ |z| \leq 1\}$ into itself has a fixed point.

1. Let $f : [0,1] \longrightarrow [0,1]$ be a continuous map. Show that $f$ has a fixed point. Can you prove the analogous statement for the square $[0,1] \times [0,1]$?

2. Let $f : \overline{D} \longrightarrow \mathbb{S}^1$ be a continuous map. Show that there exists a continuous function $\varphi : \overline{D} \longrightarrow \mathbb{R}$ such that $f = e^{i\varphi}$.

3. Let $g : \mathbb{S}^1 \longrightarrow \mathbb{R}$ be continuous. Show that there exists $x \in \mathbb{S}^1$ such that $g(x) = g(-x)$.

4. Let $f : \overline{D} \longrightarrow \mathbb{S}^1$ be continuous. Show that there exists $z \in \mathbb{S}^1$ such that $f(z) \neq z$.

5. Let $h : \overline{D} \longrightarrow \overline{D}$ be continuous. Prove that $h$ has a fixed point.

   Hint. Assume that $h$ has no fixed point. For each $z \in \overline{D}$, let $r(z)$ be the unique intersection of the ray starting at $h(z)$ and passing through $z$ with the boundary $\mathbb{S}^1 = \partial\overline{D}$. Show that $r : \overline{D} \to \mathbb{S}^1$ is continuous and derive a contradiction using the previous question.

6. Does Brouwer's fixed point theorem remain valid in higher dimensions?