Problem 1 (yau-contest 2021)
Section: Algebra and Number Theory

For any prime $p$ and a nonzero element $a \in \mathbb{F}_p$, prove that the polynomial $A(x) = x^p - x - a$ is irreducible and separable over $\mathbb{F}_p$.

Problem 2 (yau-contest 2021)
Section: Algebra and Number Theory

Determine the automorphism group of the splitting field of $f(x) = x^3 - 3x + 1$ over $\mathbb{Q}$.

Problem 3 (yau-contest 2021)
Section: Algebra and Number Theory

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

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

(b) If $t$ denotes the element $x/y$ in the fraction field $K$ of $R$, prove that $K$ is equal to $F(t)$.

(c) Prove that $F[t]$ is the integral closure of $R$ in $K = F[t]$.

Problem 4 (yau-contest 2021)
Section: Algebra and Number Theory

Let $p_1,\ldots,p_n$ be $n$ distinct prime numbers. Show: $\sqrt{p_1}+\cdots+\sqrt{p_n}$ is not rational.

Problem 5 (yau-contest 2021)
Section: Algebra and Number Theory

Find all integral solutions $(x,y)$ for the equation $x^2 + 13 = y^3$. (Hint: You can use the fact that $\mathbb{Q}(\sqrt{-13})$ has class number $2$.)

Problem 6 (yau-contest 2021)
Section: Algebra and Number Theory

Let $p$ be a prime number and $\mathbb{Q}_p$ be the field of $p$-adic numbers. Fix an algebraic closure $\overline{\mathbb{Q}}_p$ of $\mathbb{Q}_p$. Let $g : \mathbb{Z}_{\ge 0} \to \mathbb{N}$ be a strictly increasing function. For each $i \in \mathbb{Z}_{\ge 0}$, pick a primitive $(p^{g(i)} - 1)$-th root of unity $\zeta_i$ in $\overline{\mathbb{Q}}_p$.

(a) Show that for each $i \ge 0$, $K_i := \mathbb{Q}_p(\zeta_i)$ is an unramified Galois extension of $\mathbb{Q}_p$ of degree $g(i)$.

(b) Give an explicit function $g$ as above such that $K_{i-1} \subset K_i$ for all $i > 0$. Let $0 = N_0 < N_1 < N_2 < \cdots$ be an increasing sequence of nonnegative integers. Let $a_i := \sum_{j=0}^i s_j p^{N_j}$. Show that for each $i \ge 0$, $K_i = \mathbb{Q}_p(a_i)$ and that $(a_i)$ is a Cauchy sequence in $\mathbb{Q}_p$.

(c) Let $\eta \in \overline{\mathbb{Q}}_p$ be of degree $g$ over $\mathbb{Q}_p$, prove that there exists $M \in \mathbb{N}$ such that $\zeta_i$ does not satisfy any congruence
$$s_{g-1} \eta^{p^{g-1}} + s_{g-2} \eta^{p^{g-2}} + \cdots + s_1 \eta + s_0 \equiv 0 \pmod{p^M}$$
in which the $s_i$’s are $p$-adic integers not all of which are divisible by $p$.

(d) Take a suitable sequence $(N_i)$ as above such that $(a_i)$ does not converge in $\overline{\mathbb{Q}}_p$. Conclude that $\mathbb{Q}_p$ is not complete with respect to the $p$-adic topology.