Problem 1 (yau-contest 2017)
Section: Algebra and Number Theory

Let $\mathbb Q_p$ denote the field of $p$-adic numbers and let $\mathbb Z_p$ denote the ring of $p$-adic integers ($p$ is a prime number).

(a) Show that for every integer $k \ge 0$, $\,(p^{-k}\mathbb Z_p)/\mathbb Z_p \cong \mathbb Z/p^k\mathbb Z$ as abelian groups.

(b) Determine the endomorphism ring of the abelian group $(p^{-k}\mathbb Z_p)/\mathbb Z_p$ ($k \ge 0$).

(c) Determine the endomorphism ring of the abelian group $\mathbb Q_p/\mathbb Z_p$.

(d) Determine the endomorphism ring of the abelian group $\mathbb Q/\mathbb Z$.

Problem 2 (yau-contest 2017)
Section: Algebra and Number Theory

Let $A$ be a finite abelian group and let $\phi : A \to A$ be an endomorphism. Put
\[A_{\mathrm{nil}} := \{x \in A \mid \phi^{k}(x)=0 \text{ for some } k \ge 1\} .\]
(a) Show that there is a subgroup $A_0$ of $A$ such that $\phi$ restricts to an automorphism of $A_0$ and $A = A_0 \oplus A_{\mathrm{nil}}$.

(b) Show that such a subgroup is unique.

Problem 3 (yau-contest 2017)
Section: Algebra and Number Theory

Let $L/F$ be a Galois field extension, not necessarily finite. Let $x \in L$.

(a) Show that the set $\mathcal P$ of subextensions of $L/F$ not containing $x$ has a maximal element $E$. Let $K/E$ be a nontrivial finite extension contained in $L$. Show that $x \in K$.

(b) Let $K'$ be the Galois closure of $K/E$ in $L$. Show that there exists $g \in G = \operatorname{Gal}(K'/E)$ such that $g x \ne x$.

(c) Deduce that $K/E$ is a cyclic Galois extension.

Problem 4 (yau-contest 2017)
Section: Algebra and Number Theory

The goal of this problem is to prove the Chevalley–Warning theorem. Let $p$ be a prime number and $q$ a power of $p$.

(a) Let $0 \le a < q-1$ be an integer. Show that $S(X^a) := \sum_{x \in \mathbb F_q} x^a$ equals $0$. (Here we adopt the convention $x^0 = 1$ in $\mathbb F_q$ even for $x=0$.)

(b) Let $f_1,\ldots,f_m \in \mathbb F_q[X_1,\ldots,X_n]$ be polynomials in $n$ variables satisfying $\sum_{i=1}^m \deg(f_i) < n$. Show that
\[P = \prod_{i=1}^m (1 - f_i^{\,q-1})\]
satisfies
\[S(P) := \sum_{(x_1,\ldots,x_n) \in \mathbb F_q^n} P(x_1,\ldots,x_n) = 0 .\]
Deduce that $p$ divides the cardinality of the set
\[V = \{(x_1,\ldots,x_n) \in \mathbb F_q^{\,n} \mid f_i(x_1,\ldots,x_n) = 0 \ \text{for} \ \forall i\} .\]

Problem 5 (yau-contest 2017)
Section: Algebra and Number Theory

In this problem, all matrices are $n\times n$ with complex entries. Let $U$ and $V$ be matrices such that $UV \ne VU$. Assume that $U$ is diagonalizable and commutes with $VUV^{-1}$.

(a) For $\lambda, \mu \in \mathbb C$, let
\[E_{\lambda,\mu} = \{x \in \mathbb C^n \mid Ux = \lambda x,\; VUV^{-1}x = \mu x\} .\]
Show that there exist couples $(\lambda_1,\mu_1) \ne (\lambda_2,\mu_2)$, satisfying $\lambda_i \ne \mu_i$ and $E_{\lambda_i,\mu_i} \ne 0$ for $i=1,2$.

(b) For a matrix $A$, we define $\mathrm N(A) := \operatorname{tr}(A^*A)$, where $A^* = \overline{A}^{\,T}$ is the conjugate transpose of $A$. Assume that $U$ and $V$ are unitary (namely, $U^*U = V^*V$ is the identity matrix). Deduce that $\mathrm N(1+V) \ge 4$.