Problem 1 (yau-contest 2018)
Section: Algebra and Number Theory

Problem 1 (20 points).

(a) (6 points) Show that if $2^k - 1$ is a prime for some integer $k \ge 1$, then $k$ is a prime.

(b) (6 points) Show that if $2^k + 1$ is a prime for some integer $k \ge 1$, then $k$ is a power of 2.

(c) (8 points) Prove the following theorem of Goldbach: for integers $i, j \ge 0$ with $i \ne j$, the integers $2^{2^i} + 1$ and $2^{2^j} + 1$ are coprime.

Problem 2 (yau-contest 2018)
Section: Algebra and Number Theory

Problem 2 (20 points). Let $K = \mathbb{Q}(\sqrt[3]{5})$ and let $L$ be the Galois closure of $K$.

(a) (6 points) Prove that $L$ has a unique subfield $M$ satisfying $[M : \mathbb{Q}] = 2$. Prove that every prime number $p \equiv 1 \pmod{3}$ splits in $M$.

(b) (6 points) Determine all prime numbers which are ramified in $L$.

(c) (8 points) Let $p \ge 7$ be a prime number. Let $f_p$ be the inertia degree of a prime ideal of the ring of integers $\mathcal{O}_L$ of $L$ above $p$. Recall that 5 is called a cubic residue mod $p$ if $x^3 \equiv 5 \pmod{p}$ has a solution in $\mathbb{F}_p$. Prove the following decomposition law in $L$.

(i) If $p \equiv 1 \pmod{3}$ and 5 is a cubic residue mod $p$, then $p$ splits completely in $L$.

(ii) If $p \equiv 1 \pmod{3}$ and 5 is not a cubic residue mod $p$, then $f_p = 3$.

(iii) If $p \equiv 2 \pmod{3}$, then 5 is a cubic residue and $f_p = 2$.

Problem 3 (yau-contest 2018)
Section: Algebra and Number Theory

Problem 3 (20 points). Prove that every group of order 99 is abelian.

Problem 4 (yau-contest 2018)
Section: Algebra and Number Theory

Problem 4 (20 points). Let $K$ be a field and let $V$ be a finite-dimensional $K$-vector space.

(a) (6 points) Assume that $K$ is infinite. Show that $V$ is not the union of finitely many proper linear $K$-subspaces.

(b) (6 points) Assume that $K$ is finite and $V$ is non-zero. Let $S$ be the set of affine hyperplanes of $V$. Let $g: V \to \mathbb{R}$ be a function. The Radon transform $Rg: S \to \mathbb{R}$ is defined by $(Rg)(\xi) = \sum_{x \in \xi} g(x)$ for $\xi \in S$. Show that $Rg = 0$ implies $g = 0$.

(c) (8 points) Let $v_1, \ldots, v_n, w_1, \ldots, w_n \in V$. Assume that for every $K$-linear map $f: V \to K$, $(f(v_1), \ldots, f(v_n))$ and $(f(w_1), \ldots, f(w_n))$ coincide up to permutation of the indices. Deduce that $(v_1, \ldots, v_n)$ and $(w_1, \ldots, w_n)$ coincide up to permutation of the indices. Here we make no assumptions on $K$.

Problem 5 (yau-contest 2018)
Section: Algebra and Number Theory

Problem 5 (20 points). Let $p$ be a prime number and let $v_p(\cdot)$ denote the $p$-adic valuation on $\mathbb{Q}_p$. Let $A = (a_{ij})_{1 \le i,j \le n} \in M_n(\mathbb{Q}_p)$ be an $n \times n$ matrix with entries in $\mathbb{Q}_p$. Assume that we know the following:

(1) $A^2 = p^{n+1} \cdot I_{n \times n}$;

(2) $v_p(a_{ij}) \ge i$ for all $i,j$.

Prove that $v_p(a_{ij}) \ge \max\{i, n+1-j\}$ and $a_{i,\,n+1-i} \in p^i \mathbb{Z}_p^{\times}$, i.e., $A$ has entries of the form $p^{\max\{i, n+1-j\}}$ times a $p$-adic integer with units on the antidiagonal.

Hint. Consider the antidiagonal matrix
$$
J = \begin{pmatrix}
0 & 0 & \cdots & 0 & p \\
0 & 0 & \cdots & p^2 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & p^{n-1} & \cdots & 0 & 0 \\
p^n & 0 & \cdots & 0 & 0
\end{pmatrix}.
$$