Problem Q. 1 (qiuzhen-qualify 2025-fall)
Section: Basic Questions

For $n \ge 1$, let $A, B$ be two $n \times n$ matrices over an algebraically closed field $K$ such that $AB = BA$. Prove that $A$ and $B$ have a nonzero common eigenvector.

Problem Q. 2 (qiuzhen-qualify 2025-fall)
Section: Basic Questions

Let $R = K[x,y,z,w]$ denote the polynomial ring in four indeterminates over a field $K$. Define the ideal $I \subset R$ by

$$I = \{\, f(x,y,z,w) \in R \mid f(s^3, s^2 t, s t^2, t^3) = 0 \text{ in } K[s,t] \,\}.$$

Show that $I$ cannot be generated by two elements as an ideal of $R$.

Problem Q. 3 (qiuzhen-qualify 2025-fall)
Section: Basic Questions

Let $f(x) \in \mathbb{Q}[x]$ be an irreducible polynomial of degree $4$ and assume that $f$ has exactly two real roots. Write $G$ for the Galois group of the splitting field $K$ of $f$ over $\mathbb{Q}$.

1. Show that $8$ divides the order $|G|$.
2. Show that $G$ is not abelian.

Problem Q. 4 (qiuzhen-qualify 2025-fall)
Section: Basic Questions

Let $R$ be a ring and let $s \in R$. Set $S = \{1, s, s^2, \ldots\}$. Show that the following statements are equivalent:

1. The canonical ring morphism $R \to S^{-1}R$ is surjective.
2. There exists $m > 0$ such that $s^m R = s^M R$ for all $M \ge m$.
3. There exists $n > 0$ such that the ideal $s^n R$ is generated by an element $e$ with $e^2 = e$.

Problem Q. 5 (qiuzhen-qualify 2025-fall)
Section: Basic Questions

Let $A = \mathbb{C}[x,y]$ and $B = \mathbb{C}[u,v]$ be polynomial rings over $\mathbb{C}$, and define a $\mathbb{C}$-algebra homomorphism $f: A \to B$ by $f(x) = u$ and $f(y) = uv$. Prove or disprove each of the following statements.

1. $B$ is a flat $A$-module.
2. $B$ is an injective $A$-module.

Problem Q. 6 (qiuzhen-qualify 2025-fall)
Section: Basic Questions

1. Let $A$ be a finite abelian group and $\chi$ a complex character of $A$. Show that

$$\sum_{a \in A} |\chi(a)|^2 \ge |A| \cdot \chi(1).$$

2. Let $G$ be a finite group and $A$ an abelian subgroup of $G$ of index $n$. Let $\psi$ be a complex irreducible character of $G$. Show that

$$\psi(1) \le n.$$

Problem Q. 7 (qiuzhen-qualify 2025-fall)
Section: Basic Questions

Let $\mathrm{SU}_2$ be the Lie group of $2 \times 2$ special unitary matrices and $\mathfrak{su}_2$ the Lie algebra of $\mathrm{SU}_2$. Prove that the image of the adjoint map $\mathrm{Ad}: \mathrm{SU}_2 \to \mathrm{GL}(\mathfrak{su}_2)$ is a Lie group isomorphic to $\mathrm{SO}(3, \mathbb{R})$.

Problem Q. 8 (qiuzhen-qualify 2025-fall)
Section: Advanced Questions

Suppose $r \le n$ are positive integers. Let $X$ be the subset of $M(n \times n, \mathbb{C})$ consisting of complex $n \times n$ matrices with rank at most $r$. Prove that $X$ is Zariski closed. Find the number of irreducible components of $X$, and calculate the Krull dimension of each irreducible component of $X$.

Problem Q. 9 (qiuzhen-qualify 2025-fall)
Section: Advanced Questions

Let $C$ be a genus one complex smooth projective curve. Take any point $p \in C$.

1. Prove that there exist rational functions $f, g$ on $C$ such that $f$ has pole of order $2$ at $p$, and $g$ has pole of order $3$ at $p$, and both $f, g$ are regular on $C - \{p\}$.

Problem unknown (qiuzhen-qualify 2025-fall)

Prove that $C$ is isomorphic to a cubic curve in the complex projective plane $CP^2$.

Problem Q. 10 (qiuzhen-qualify 2025-fall)

Let $K = \mathbf{Q}(\sqrt{82})$.

1. Find the absolute value $|\operatorname{disc}(K)|$ of the discriminant $\operatorname{disc}(K)$ of $K$.

2. Determine the class group $\operatorname{Cl}(K)$ of $K$. Give a presentation of $\operatorname{Cl}(K)$ (i.e., a set of generators and relations).

Problem Q. 11 (qiuzhen-qualify 2025-fall)

Let $\mathbf{Q}_p$ denote the field of $p$-adic numbers for a prime $p$ and fix an algebraic closure $\overline{\mathbf{Q}}_p$.

1. Take $\alpha \in \overline{\mathbf{Q}}_p$, satisfying $\alpha^{p-1} = -p$ and set $K := \mathbf{Q}_p(\alpha) \subset \overline{\mathbf{Q}}_p$. Prove that $K$ is Galois over $\mathbf{Q}_p$.

2. Take $\zeta \in \overline{\mathbf{Q}}_p$ satisfying $\zeta^p = 1$ and $\zeta \ne 1$, and set $L := \mathbf{Q}_p(\zeta) \subset \overline{\mathbf{Q}}_p$. Prove $L = K$.

Problem Q. 12 (qiuzhen-qualify 2025-fall)

Consider the category $\mathcal{O}$ for $\mathfrak{sl}_2$. We identify the weight lattice of $\mathfrak{sl}_2$ with $\mathbf{Z}$ by sending the first fundamental weight to $1$. We denote by $L(\mu)$ the simple $\mathfrak{sl}_2$-module of highest weight $\mu$. In particular, the trivial representation $\mathbf{C} = L(0)$. Let $M(\mu)$ be the Verma module of highest weight $\mu$. Let $P(\mu)$ be the projective cover of $L(\mu)$.

1. Show that there is only one $\lambda \ne 0$ such that $L(\lambda)$ has the same central character as $L(0)$. Determine the value of $\lambda$.

2. Write down composition series for the Verma modules $M(0)$ and $M(\lambda)$ for $\lambda$ above and justify your answer.

3. Write down composition series for the projective modules $P(0)$ and $P(\lambda)$ for $\lambda$ above and justify your answer.

Problem Q. 13 (qiuzhen-qualify 2025-fall)

Consider the flag variety for $GL_3$:

$$\mathcal{B} = \{V_\bullet = (0 = V_0 \subset V_1 \subset V_2 \subset V_3 = \mathbf{C}^3) \mid \dim(V_i) = i,\; 1 \le i \le 3\}.$$

Let

$$e = \begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}.$$

Let $\mathcal{B}_e$ be the closed subvariety in $\mathcal{B}$ consisting of flags stabilized by $e$, that is,

$$\mathcal{B}_e = \{V_\bullet \in \mathcal{B} \mid e(V_i) \subset V_i,\; 1 \le i \le 3\}.$$

1. Compute the irreducible components of $\mathcal{B}_e$.

2. Compute the singular cohomology of $\mathcal{B}_e$ with coefficient in $\mathbf{Q}$ as a $\mathbf{Q}$-vector space.