Problem 1 (yau-contest 2012)
Section: Algebra and Number Theory

Prove that the polynomial $x^6 + 30x^5 - 15x^3 + 6x - 120$ cannot be written as a product of two polynomials of rational coefficients and positive degrees.

Problem 2 (yau-contest 2012)
Section: Algebra and Number Theory

Let $\mathbb{F}_p$ be the field of $p$-elements and $GL_n(\mathbb{F}_p)$ the group of invertible $n\times n$ matrices.

(1) Compute the order of $GL_n(\mathbb{F}_p)$.

(2) Find a Sylow $p$-subgroup of $GL_n(\mathbb{F}_p)$.

(3) Compute the number of Sylow $p$-subgroups.

Problem 3 (yau-contest 2012)
Section: Algebra and Number Theory

Let $V$ be a finite dimensional vector space over complex field $\mathbb{C}$ with a nondegenerate symmetric bilinear form $(\, , \,)$. Let
$$O(V)=\{g\in GL(V)\mid (gu,gv)=(u,v),\ u,v\in V\}$$
be the orthogonal group. Prove that the fixed point subspace $(V\otimes_{\mathbb{C}} V)^{O(V)}$ is 1-dimensional.

Problem 4 (yau-contest 2012)
Section: Algebra and Number Theory

Let $\mathcal{D}$ be the ring consisting of all linear differential operators of finite order on $\mathbb{R}$ with polynomial coefficients, of the form
$$D=\sum_{i=0}^{n} a_i(x)\, \frac{d^i}{dx^i}$$
for some natural number $n\in\mathbb{N}$ and polynomials $a_0(x),\dots,a_n(x)\in \mathbb{R}[x]$. This ring $\mathcal{D}$ operates naturally on $M:=\mathbb{R}[x]$, making $M$ a left $\mathcal{D}$-module.

(1) (to warm up) Suppose that $b(x)\in \mathbb{R}[x]$ is a non-zero polynomial in $M$, and let $c(x)$ be any element in $M$. Show that there is an element $D\in \mathcal{D}$ such that $D(b(x))=c(x)$.

(2) Suppose that $m$ is a positive integer, $b_1(x),\dots,b_m(x)$ are $m$ polynomials in $M$ linearly independent over $\mathbb{R}$ and $c_1(x),\dots,c_m(x)$ are $m$ polynomials in $M$. Prove that there exists an element $D\in \mathcal{D}$ such that $D(b_i(x))=c_i(x)$ for $i=1,\dots,m$.

Problem 5 (yau-contest 2012)
Section: Algebra and Number Theory

Let $\Lambda$ be a lattice of $\mathbb{C}$, i.e., a subgroup generated by two $\mathbb{R}$-linear independent elements. Let $R$ be the subring of $\mathbb{C}$ consists of elements $\alpha$ such that $\alpha\Lambda\subset \Lambda$. Let $R^{\times}$ denote the group of invertible elements in $R$.

(1) Show that either $R=\mathbb{Z}$ or have rank $2$ over $\mathbb{Z}$.

(2) Let $n\ge 3$ be a positive integer and $(R/nR)^{\times}$ the group of invertible elements in the quotient $R/nR$. Show that the canonical group homomorphism
$$R^{\times}\to (R/nR)^{\times}$$
is injective.

(3) Find maximal size of $R^{\times}$.

Problem 6 (yau-contest 2012)
Section: Algebra and Number Theory

Let $V$ be a (possible) infinite dimensional vector space over $\mathbb{R}$ with a positive definite quadratic norm $\|\cdot\|$. Let $A$ be an additive subgroup of $V$ with following properties:

(1) $A/2A$ is finite;

(2) for any real number $c$ the set
$$\{a\in A:\ \|a\|<c\}$$
is finite.

Prove that $A$ is of finite rank over $\mathbb{Z}$.