Problem 1 (yau-contest 2020)
Section: Algebra and Number Theory

Let $F$ be a field of characteristic zero. Consider the polynomial ring $F[x_1,\ldots,x_n]$.

(a) Prove Newton’s identity over the field $F$

$$p_k - p_{k-1} e_1 + \cdots + (-1)^{k-1} p_1 e_{k-1} + (-1)^k k e_k = 0,$$

where

$$e_k(x_1,\ldots,x_n) = \sum_{1 \le i_1 < \cdots < i_k \le n} x_{i_1}\cdots x_{i_k}$$

for $1 \le k \le n$, $e_0 = 1$, $e_k = 0$ when $k > n$, and

$$p_k(x_1,\ldots,x_n) = x_1^k + \cdots + x_n^k.$$

(b) Prove that over the field $F$ of characteristic zero, an $n \times n$ matrix $A$ is nilpotent if and only if the trace of $A^k$ is equal to zero for all $k = 1,2,\ldots$.

Hint: use Part (a).

(c) Prove that over the field of $F$ of characteristic zero, two $n \times n$ matrix $A$ and $B$ have the same characteristic polynomial if and only if the trace of $A^k$ and trace of $B^k$ are equal for all $k = 1, 2, \ldots$.

Hint: use Part (a).

Problem 2 (yau-contest 2020)
Section: Algebra and Number Theory

(a) Let $M$ be a finitely generated $R$-module and $a \subset R$ an ideal. Suppose $\phi : M \to M$ is an $R$-module map such that $\phi(M) \subset aM$. Prove that there is a monic polynomial $p(t) \in R[t]$ with coefficients from $a$ such that $p(\phi) = 0.

Hint: $p(t)$ is basically just the characteristic polynomial.

(b) If $M$ is a finitely generated $R$-module such that $aM = M$ for some ideal $a \subset R$, then there exits $x \in R$ such that $1 - x \in a$ and $xM = 0.

Problem 3 (yau-contest 2020)
Section: Algebra and Number Theory

Let $R = F[x,y]/(y^2 - x^2 - x^3)$ for some field $F$.

(a) Prove that $R$ is an integral domain.

(b) Compute the normalization of $R$ (i.e., the integral closure of $R$ in its field of fraction).

Problem 4 (yau-contest 2020)
Section: Algebra and Number Theory

Let $p$ and $\ell$ be two prime numbers and $[\ell x]$ denote the $\ell$-th cyclotomic polynomial $1 + x + \cdots + x^{\ell-1}$.

(a) Prove that $[\ell x]$ is an irreducible element of $\mathbb{Q}[x]$.

(b) Show that $[\ell x]$ is divisible by $x - 1$ in $\mathbb{F}_p[x]$ if $p \equiv \ell$. Here $\mathbb{F}_p$ is the finite field $\mathbb{Z}/p\mathbb{Z}$.

(c) Suppose $p \ne \ell$, let $a$ be the order of $p$ in $\mathbb{F}_\ell^\times$. Show that $a$ is the first value of $m$ for which the group $\mathrm{GL}_m(\mathbb{F}_p)$ of invertible $m \times m$ matrices with entries from $\mathbb{F}_p$ contains an element of order $\ell$.

Hint: Derive and use the formula for the number of elements in $\mathrm{GL}_m(\mathbb{F}_p)$.

Problem 5 (yau-contest 2020)
Section: Algebra and Number Theory

Let $p \ge 3$ be a prime number and let $\mathbb{Z}_p$ be the ring of $p$-adic integers.

(a) Show that an element in $1 + p\mathbb{Z}_p$ is a $p$-th power in $\mathbb{Z}_p$ if and only if it lives in $1 + p^2 \mathbb{Z}_p$.

(b) Let $\mathbb{Z}_p^{\times}$ denote the group of units in $\mathbb{Z}_p$. Show that there exist $a,b,c \in \mathbb{Z}_p^{\times}$ such that $a^p + b^p = c^p$ if and only if

$$\sum_{i=1}^{p-1} i^{p-2} i^t \equiv 0 \pmod p$$

for some integer $t \in \{2,3,\ldots,p-1\}$. (In particular, this condition holds for $p = 7$ by taking $t = 3$. Therefore, Fermat’s Last Theorem does not hold for $\mathbb{Z}_7$.)

Problem 6 (yau-contest 2020)
Section: Algebra and Number Theory

Recall that a metric space is called spherically complete if any decreasing sequence of closed balls has nonempty intersection.

Let $p$ be a prime number and let $\mathbb{Q}_p$ be the field of $p$-adic numbers. For every integer $n \ge 1$, consider the finite extension $\mathbb{Q}_p(\mu_{p^n})$ of $\mathbb{Q}_p$ generated by all $p^n$-th roots of unity. Let $\mathbb{Q}_p(\mu_{p^{\infty}}) = \bigcup_{n \ge 1} \mathbb{Q}_p(\mu_{p^n})$. All of these algebraic extensions of $\mathbb{Q}_p$ are equipped with the unique norm $|\cdot|$ extending the usual $p$-adic norm on $\mathbb{Q}_p$.

Question: Which of the following are spherically complete? Explain why.

(a) $\mathbb{Q}_p$;

(b) $\mathbb{Q}_p(\mu_{p^n})$;

(c) $\mathbb{Q}_p(\mu_{p^{\infty}})$;

(d) $\overline{\mathbb{Q}_p(\mu_{p^{\infty}})}$, the completion of $\mathbb{Q}_p(\mu_{p^{\infty}})$.

Hint: Show that there exists a sequence $a_1,a_2,\ldots \in \overline{\mathbb{Q}_p(\mu_{p^{\infty}})}$ such that $|a_1| > |a_2| > \cdots$ and $\lim |a_i| > 0$, and such that the closed balls

$$B_i := \left\{ x \in \mathbb{Q}_p(\mu_{p^{\infty}}) : |x - a_1 - a_2 - \cdots - a_i| \le |a_i| \right\}$$

have empty intersection.