Problem 1 (yau-contest 2019)
Section: Algebra and Number Theory

Let $G$ be a finite group. Assume that for any representation $V$ of $G$ over a field of characteristic zero, the character $\chi_V$ takes value in $\mathbb{Q}$. Assume $g$ is an element in $G$ such that $g^{2019} = 1$. Prove that $g$ and $g^{19}$ are conjugate in $G$.

Problem 2 (yau-contest 2019)
Section: Algebra and Number Theory

Let $p$ be a prime number, and let $\mathbb{F}_p$ be the finite field with $p$ elements. Let $F = \mathbb{F}_p(t)$ be the field of rational functions over $\mathbb{F}_p$. Consider all subfields $C$ of $F$ such that $F/C$ is a finite Galois extension.

1. Show that among such subfields, there is a smallest one $C_0$, i.e., $C_0$ is contained in any other $C$.
2. What is the degree of $F/C_0$?

Problem 3 (yau-contest 2019)
Section: Algebra and Number Theory

Let $R \subset R'$ be an integral extension of commutative rings. Let $\mathfrak{p}'$ be a prime ideal of $R'$. Prove that $\mathfrak{p}'$ is a maximal ideal of $R'$ if and only if $\mathfrak{p}' \cap R$ is a maximal ideal of $R$.

Problem 4 (yau-contest 2019)
Section: Algebra and Number Theory

1. Prove that $\mathrm{GL}_n(\mathbb{C})$ is path-connected.

2. Let
$$
X = \{ A \in \mathrm{GL}_n(\mathbb{C}) \mid A^m = \mathrm{Id} \},
$$
Describe the path-connected component of $X$ and prove your answer.

Problem 5 (yau-contest 2019)
Section: Algebra and Number Theory

The Fibonacci sequence is defined by
$$
F_0 = 0, \quad F_1 = 1, \quad F_{n+2} = F_{n+1} + F_n.
$$
Let $p$ be a prime number.

1. Show that if $p \equiv 1, 4 \pmod{5}$, then $p$ divides $F_{p-1}$.
2. Let $\mathbb{F}_{p^2}$ be the finite field of $p^2$ elements. Show that the norm map $N : \mathbb{F}_{p^2}^\times \to \mathbb{F}_p^\times$ is surjective, and deduce the cardinality of the kernel of $N$.
3. Show that if $p \equiv 2, 3 \pmod{5}$, then $p$ divides $F_{p+1}$.