Problem 1 (yau-contest 2011)
Section: Algebra, Number Theory and Combinatorics

Let $F$ be a field and $\hat{F}$ the algebraic closure of $F$. Let $f(x,y)$ and $g(x,y)$ be polynomials in $F[x,y]$ such that $\gcd(f,g)=1$ in $F[x,y]$. Show that there are only finitely many $(a,b)\in \hat{F}^2$ such that $f(a,b)=g(a,b)=0$. Can you generalize this to the cases of more than two variables?

Problem 2 (yau-contest 2011)
Section: Algebra, Number Theory and Combinatorics

Let $D$ be a PID, and $D^n$ the free module of rank $n$ over $D$. Then any submodule of $D^n$ is a free module of rank $m \le n$.

Problem 3 (yau-contest 2011)
Section: Algebra, Number Theory and Combinatorics

Identify pairs of integers $n\ne m\in \mathbb{Z}_+$ such that the quotient rings $\mathbb{Z}[x,y]/(x^2-y^n) \cong \mathbb{Z}[x,y]/(x^2-y^m)$; and identify pairs of integers $n\ne m\in \mathbb{Z}_+$ such that $\mathbb{Z}[x,y]/(x^2-y^n) \not\cong \mathbb{Z}[x,y]/(x^2-y^m)$.

Problem 4 (yau-contest 2011)
Section: Algebra, Number Theory and Combinatorics

Is it possible to find an integer $n>1$ such that the sum $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n}$$ is an integer?

Problem 5 (yau-contest 2011)
Section: Algebra, Number Theory and Combinatorics

Recall that $\mathbb{F}_7$ is the finite field with $7$ elements, and $GL_3(\mathbb{F}_7)$ is the group of all invertible $3\times 3$ matrices with entries in $\mathbb{F}_7$.
(a) Find a $7$-Sylow subgroup $P_7$ of $GL_3(\mathbb{F}_7)$.
(b) Determine the normalizer subgroup $N$ of the $7$-Sylow subgroup you found in (a).
(c) Find a $2$-Sylow subgroup of $GL_3(\mathbb{F}_7)$.

Problem 6 (yau-contest 2011)
Section: Algebra, Number Theory and Combinatorics

For a ring $R$, let $SL_2(R)$ denote the group of invertible $2\times 2$ matrices. Show that $SL_2(\mathbb{Z})$ is generated by $$T=\begin{pmatrix}1 & 1\\ 0 & 1\end{pmatrix} \quad \text{and} \quad S=\begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}.$$ What about $SL_2(\mathbb{R})$?