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

Let $a_1, \cdots, a_n$ and $b_1, \cdots, b_n$ be complex numbers such that $a_i+b_j \neq 0$ for all $i,j=1,\cdots,n$. Define $c_{ij}:=\frac{1}{a_i+b_j}$ for all $i,j=1,\cdots,n$, and let $C$ be the $n\times n$ matrix with entries $c_{ij}$. Prove that

$$\det(C)=\frac{\prod_{1\le i<j\le n}(a_i-a_j)(b_i-b_j)}{\prod_{1\le i,j\le n}(a_i+b_j)}.$$

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

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 $\mathbb{F}_7$.

(1) Find a $7$-Sylow subgroup $P_7$ of $GL_3(\mathbb{F}_7)$.

(2) Determine the normalizer subgroup $\mathcal{N}$ of the $7$-Sylow subgroup you found in (a).

(3) Find a $2$-Sylow subgroup of $GL_3(\mathbb{F}_7)$.

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

Let $V$ be a finite dimensional vector space with a positive definite quadratic form $(\, - , - \,)$. Let $O(V)$ denote the orthogonal group:

$$O(V)=\{\, g\in GL(V) : (g x, g y)=(x,y), \ \forall x,y\in V \,\}.$$

For any non-zero $v\in V$, let $s_v$ denote the reflection on $V$:

$$s_v(w)=w-2\,\frac{(v,w)}{(v,v)}\,v.$$

(1) Show that $s_v\in O(V)$.

(2) Show that if $v$ and $w$ are vectors in $V$ with $\|v\|=\|w\|$, then there is either a reflection or product of two reflections that takes $v$ into $w$.

(3) Deduce that every element of the orthogonal group of $V$ can be written as the product of at most $2\,\dim V$ reflections.

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

Consider the real Lie group $SL_2(\mathbb{R})$ of $2$ by $2$ matrices of determinant one. Compute the fundamental group of $SL_2(\mathbb{R})$ and describe the Lie group structure on the universal covering

$$\widetilde{SL_2(\mathbb{R})}\to SL_2(\mathbb{R}).$$

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

Let $f\in \mathbb{C}[x,y,z]$ be an irreducible homogenous polynomial of degree $d>0$. For each integer $n\ge d$, define

$$P(n)=\dim_{\mathbb{C}}\,\mathbb{C}[x,y,z]_n / f\cdot \mathbb{C}[x,y,z]_{n-d}$$

where $\mathbb{C}[x,y,z]_n$ is the subspace of homogenous polynomials of degree $n$. Show there are constants $c$ such that for $n$ sufficiently large,

$$P(n)=dn+c.$$

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

Let $p$ be an odd prime and $\mathbb{Z}_p$ the $p$-adic integer which can be defined as the projective limit of $\mathbb{Z}/p^n\mathbb{Z}$ and let $\mathbb{Q}_p$ be its fractional field. Let $\mathbb{Z}_p^\times$ denote the group of invertible elements in $\mathbb{Z}_p$ which is also the projective limit of $(\mathbb{Z}/p^n\mathbb{Z})^\times$.

(1) For any integer $a$ is not divisible by $p$, show that the sequence $(a^{p^n})_n$ convergent to an element $\omega(a)\in \mathbb{Z}_p$ satisfying

$$\omega(a)^{p-1}=1,\qquad \omega(a)\equiv a \pmod{p}.$$

Moreover, $\omega(a)$ depends only on $a \bmod p$.

(2) Define a logarithmic function $\log$ on $1+p\mathbb{Z}_p$ by usual formula:

$$\log(1+px)=\sum_{n=1}^\infty (-1)^{n-1}\frac{p^n}{n}x^n.$$

Show that the logarithmic function is convergent and define an isomorphism

$$1+p\mathbb{Z}_p \to p\mathbb{Z}_p.$$

Moreover, on the dense subgroup $\log(1+p)\mathbb{Z}$, the inverse is given by

$$\log(1+p)\cdot x \mapsto (1+p)^x,\qquad \forall x\in \mathbb{Z}.$$

(3) Deduce from above that $\mathbb{Z}_p^\times \simeq \mathbb{Z}_p \times \mathbb{Z}/(p-1)\mathbb{Z}$.