Problem 1 (yau-contest 2015)
Section: Linear Algebra

Let $V = \mathbb{R}^n$ be an Euclidean space equipped with usual inner product, and $g$ an orthogonal matrix acting on $V$. For $a \in V$, let $s_a$ denote the reflection

$$s_a(x) := x - 2 \frac{(x,a)}{(a,a)} a, \quad \forall x \in V.$$

(1.1) For $a = (g-1)b \ne 0$, show that

$$\ker(s_a g - 1) = \ker(g - 1) \oplus \mathbb{R} b.$$

(1.2) Show that $g$ is a product of $\dim((g - 1)V)$ reflections.

Problem 2 (yau-contest 2015)
Section: Group Theory

Let $p$ and $q$ be two distinct prime numbers. Let $G$ be a non-abelian finite group satisfying the following conditions:

1. all nontrivial elements have order either $p$ or $q$;
2. the $q$-Sylow subgroup $H_q$ is normal and is a nontrivial abelian group.

Show in steps the following statement:

The group $G$ is of the form $(\mathbb{Z}/p\mathbb{Z}) \times (\mathbb{Z}/q\mathbb{Z})^n$, where the action of $1 \in \mathbb{Z}/p\mathbb{Z}$ on $(\mathbb{Z}/q\mathbb{Z})^n \simeq \mathbb{F}_q^n$ is given by a matrix $M(1) \in \mathrm{GL}_n(\mathbb{F}_q)$ whose eigenvalues are all primitive $p$-th roots of unities.

(2.1) Let $H_p$ denote a $p$-Sylow subgroup of $G$. Show that its inclusion into $G$ induces an isomorphism $H_p \cong G/H_q$, and that $G \simeq H_p \ltimes H_q$.

(2.2) Let $M : H_p \to \mathrm{Aut}(H_q) \simeq \mathrm{GL}_n(\mathbb{F}_q)$ be the homomorphism induced from the conjugations. Show that for each $1 \ne a \in H_p$, $M(a)$ is semisimple whose eigenvalues are all primitive $p$-th roots of unities. In particular $M$ is injective.

(2.3) Show that if two nontrivial elements $a, b \in H_p$ commute with each other, then $a = b^n$ for some $n \in \mathbb{N}$, and that $H_p \simeq \mathbb{Z}/p\mathbb{Z}$.

(2.4) Complete the solution of the problem.

Problem 3 (yau-contest 2015)
Section: Number Theory

Let $\zeta$ be a root of unity satisfying an equation $\zeta = 1 + N\eta$ for an integer $N \ge 3$ and an algebraic integer $\eta$. Show that $\zeta = 1$.

Problem 4 (yau-contest 2015)
Section: Representation Theory

Let $G$ be a finite group and $(\pi, V)$ a finite dimensional $\mathbb{C}G$-module. For $n \ge 0$, let $\mathbb{C}[V]_n$ be the space of homogeneous polynomial functions on $V$ of degree $n$. For a simple $G$-representation $\rho$, denote by $a_n(\rho)$ the multiplicity of $\rho$ in $\mathbb{C}[V]_n$. Show that

$$\sum_{n\ge 0} a_n(\rho) t^n = \frac{1}{|G|} \sum_{g\in G} \frac{\chi_\rho(g)}{\det(\mathrm{id}_V - \pi(g) t)}.$$

Problem 5 (yau-contest 2015)
Section: Linear Algebra

Let $A$ be an $n \times n$ complex matrix considered as an operator on $V = (\mathbb{C}^n, (\cdot,\cdot))$ with standard hermitian form. Let $A^* = \bar{A}^t$ be the hermitian transpose of $A$:

$$(Ax, y) = (x, A^* y), \quad \forall x, y \in \mathbb{C}^n.$$

(5.1) For any $\lambda \in \mathbb{C}$, show the identity:

$$\ker(A - \lambda)^\perp = (A^* - \bar{\lambda}) V.$$

(5.2) Show the equivalence of the following two statements:

(a) $A$ commutes with $A^*$;

(b) there is a unitary matrix $U$ (in the sense $U^* = U^{-1}$), such that $UAU^{-1}$ is diagonal.

Problem 6 (yau-contest 2015)
Section: Galois Theory

Consider the polynomial $f(x) = x^5 - 80x + 5$.

(6.1) Show that $f$ is irreducible over $\mathbb{Q}$.

(6.2) Show in steps that the split field $K$ of $f$ has Galois group $G := \mathrm{Gal}(K/\mathbb{Q})$ isomorphic to $S_5$.

(a) $f = 0$ has exactly two complex roots;

(b) $G$ can be embedded into $S_5$ with image containing cycles $(12345)$ and $(12)$;

(c) $G \simeq S_5$.