Problem 1 (yau-contest 2010)
Section: Algebra, Number Theory and Combinatorics

Let $V$ be a finite dimensional complex vector space. Let $A, B$ be two linear endomorphisms of $V$ satisfying $AB - BA = B$. Prove that there is a common eigenvector for $A$ and $B$.

Problem 2 (yau-contest 2010)
Section: Algebra, Number Theory and Combinatorics

Let $M_2(\mathbb{R})$ be the ring of $2\times 2$ matrices with real entries. Its group of multiplicative units is $GL_2(\mathbb{R})$, consisting of invertible matrices in $M_2(\mathbb{R})$.

(a) Find an injective homomorphism from the field $\mathbb{C}$ of complex numbers into the ring $M_2(\mathbb{R})$.
(b) Show that if $\phi_1$ and $\phi_2$ are two such homomorphisms, then there exists $g \in GL_2(\mathbb{R})$ such that $\phi_2(x) = g\,\phi_1(x)\,g^{-1}$ for all $x \in \mathbb{C}$.
(c) Let $h$ be an element in $GL_2(\mathbb{R})$ whose characteristic polynomial $f(x)$ is irreducible over $\mathbb{R}$. Let $F \subset M_2(\mathbb{R})$ be the subring generated by $h$ and $a \cdot I$ for all $a \in \mathbb{R}$, where $I$ is the identity matrix. Show that $F$ is isomorphic to $\mathbb{C}$.
(d) Let $h'$ be any element in $GL_2(\mathbb{R})$ with the same characteristic polynomial $f(x)$ as $h$ in (c). Show that $h$ and $h'$ are conjugate in $GL_2(\mathbb{R})$.
(e) If $f(x)$ in (c) and (d) is reducible over $\mathbb{R}$, will the same conclusion on $h$ and $h'$ hold? Give reasons.

Problem 3 (yau-contest 2010)
Section: Algebra, Number Theory and Combinatorics

Let $G$ be a non-abelian finite group. Let $c(G)$ be the number of conjugacy classes in $G$. Define $\bar c(G) := c(G)/|G|$, ($|G| = \operatorname{Card}(G)$).

(a) Prove that $\bar c(G) \le \tfrac{5}{8}$.
(b) Is there a finite group $H$ with $\bar c(H) = \tfrac{5}{8}$?
(c) (open ended question) Suppose that there exists a prime number $p$ and an element $x \in G$ such that the cardinality of the conjugacy class of $x$ is divisible by $p$. Find a good/sharp upper bound for $\bar c(G)$.

Problem 4 (yau-contest 2010)
Section: Algebra, Number Theory and Combinatorics

Let $F$ be a splitting field over $\mathbb{Q}$ the polynomial $x^8 - 5 \in \mathbb{Q}[x]$. Recall that $F$ is the subfield of $\mathbb{C}$ generated by all roots of this polynomial.

(a) Find the degree $[F:\mathbb{Q}]$ of the number field $F$.
(b) Determine the Galois group $\operatorname{Gal}(F/\mathbb{Q})$.

Problem 5 (yau-contest 2010)
Section: Algebra, Number Theory and Combinatorics

Let $T \subset \mathbb{N}_{>0}$ be a finite set of positive integers. For each integer $n>0$, define $a_n$ to be the number of all finite sequences $(t_1,\ldots,t_m)$ with $m \le n$, $t_i \in T$ for all $i=1,\ldots,m$ and $t_1 + \cdots + t_m = n$. Prove that the infinite series
$$1 + \sum_{n \ge 1} a_n z^n \in \mathbb{C}[[z]]$$
is a rational function in $z$, and find this rational function.

Problem 6 (yau-contest 2010)
Section: Algebra, Number Theory and Combinatorics

Describe all the irreducible complex representations of the group $S_4$ (the symmetric group on four letters).