Problem 1 (qiuzhen-qualify 2024-fall)

Suppose $q$ is a prime power. Compute the number of monic irreducible polynomials of degree $21$ over $\mathbb{F}_q$.

Problem 2 (qiuzhen-qualify 2024-fall)

Given a positive integer $n$. Let $\mathfrak{g}=\mathfrak{sl}_{n+1}(\mathbb{C})$ be the Lie algebra of all $(n+1)\times(n+1)$ complex matrices with trace $0$. Let $\mathfrak{h}$ be the space of all diagonal matrices in $\mathfrak{g}$.

(i) Give the Cartan decomposition (also called root decomposition) of $\mathfrak{g}$ with respect to $\mathfrak{h}$.

(ii) Describe all roots in the root system associated with $(\mathfrak{g},\mathfrak{h})$, and prove your result.

Problem 3 (qiuzhen-qualify 2024-fall)

Let $p$ be a prime number. Consider group
$$
G=\left\{\begin{pmatrix}1&0\\ a&b\end{pmatrix}\;\middle|\; a\in \mathbb{F}_p,\; b\in \mathbb{F}_p^{\times}\right\}\subset \mathrm{GL}(2,\mathbb{F}_p)
$$
with group law given by matrix multiplication. Find the character table for irreducible complex representations of $G$.

Problem 4 (qiuzhen-qualify 2024-fall)

Suppose $G$ is a finite non-abelian simple group such that $|G|\le 60$. Find all possible $G$.

Problem 5 (qiuzhen-qualify 2024-fall)

Suppose $q$ is a prime power.

(i) Prove that any nilpotent square matrix over $\mathbb{F}_q$ is similar to a matrix of the Jordan normal form.

(ii) Compute the number of $3\times3$ nilpotent matrices over $\mathbb{F}_q$.

Problem unknown (qiuzhen-qualify 2024-fall)

Let $R$ be any commutative ring, and consider the commutative diagram (1) (see above) of $R$-module homomorphisms. Suppose we have

(i) All the columns are short exact sequences;

(ii) The first and third rows are short exact sequences;

(ii) The composition $A_2 \to B_2 \to C_2$ is zero.

Prove that the second row is a short exact sequence.

Problem 7 (qiuzhen-qualify 2024-fall)

Let $F = Q\left[\cos \frac{\pi}{15}\right]$.

(i) Prove $F/Q$ is a Galois extension.

(ii) Find the structure of $\operatorname{Gal}(F/Q)$.

(iii) Find a chain $Q = F_0 \subset F_1 \subset \cdots \subset F_k = F$ of field extensions, such that $[F_{i+1} : F_i]$ are prime numbers.

Problem 8 (qiuzhen-qualify 2024-fall)

Consider the ring $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$.

(i) Prove that $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$ is a Euclidean domain.

(ii) Let $p \ge 5$ be a prime number. Let $\alpha \in \overline{\mathbb{F}}_p \setminus \mathbb{F}_p$ be an element so that $\alpha^3 = -1$. Deduce that $(\alpha - \alpha^{-1})^2 = -3$, and show that $p$ is irreducible in $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$ if and only if $p \equiv 2 \pmod{3}$.

Problem 9 (qiuzhen-qualify 2024-fall)

Let $p$ be a prime number. Suppose $R$ is a local ring such that for any element $r \in R$, we have $r^p = r$. Let $I$ be the maximal ideal of $R$.

(i) Find all the possibilities of $R/I$;

(ii) Find all the possibilities of $R$.