Problem 1 (qiuzhen-qualify 2025-spring)
Section: Basic Questions

Let $V$ be a complex vector space of finite dimension and $T$ a linear operator on $V$. Suppose the minimal polynomial of $T$ agrees with the characteristic polynomial. Show that $T$ only has finitely many invariant subspaces.

Problem 2 (qiuzhen-qualify 2025-spring)
Section: Basic Questions

Denote by $D_{2n}$ the dihedral group of order $2n$. Prove that the normalizer of any Sylow $5$-subgroup of $A_6$ is isomorphic to $D_{10}$.

Problem 3 (qiuzhen-qualify 2025-spring)
Section: Basic Questions

1. Is $\mathbb{Q}(\sqrt{1+\sqrt{2}})/\mathbb{Q}$ a Galois extension?

2. Determine $\operatorname{Aut}(\mathbb{Q}(\sqrt{1+\sqrt{2}}))$, i.e., the automorphism group of $\mathbb{Q}(\sqrt{1+\sqrt{2}})$.

Problem 4 (qiuzhen-qualify 2025-spring)
Section: Basic Questions

1. Let $R$ be a principal ideal domain with fraction field $K$. Let $R'$ be a subring of $K$ that contains $R$. Show that $R'$ is a principal ideal domain.

2. Let $R_1$ and $R_2$ be commutative rings. Show that the ideals of $R_1 \times R_2$ are all of the form $I_1 \times I_2$, where $I_1$ and $I_2$ are ideals of $R_1$ and $R_2$, respectively.

Problem 5 (qiuzhen-qualify 2025-spring)
Section: Basic Questions

Let $M$ be a Noetherian $R$-module and $I$ the annihilator of $M$. Show that $R/I$ is Noetherian.

Problem 6 (qiuzhen-qualify 2025-spring)
Section: Basic Questions

Let $\mathfrak{sl}(n,\mathbb{C})$ be the Lie algebra of $n \times n$ traceless matrices over $\mathbb{C}$.

1. Compute the Killing form for $\mathfrak{sl}(n,\mathbb{C})$.

2. Recall the Weyl denominator formula
$$
\sum_{w\in W} (-1)^{l(w)} e^{w(\rho)} = \prod_{\alpha\in R^+} \left(e^{\alpha/2} - e^{-\alpha/2}\right),
$$
where $R^+$ denotes the positive roots and $\rho$ is the half sum of the positive roots. Prove this formula for $\mathfrak{sl}(n,\mathbb{C})$.

Problem 7 (qiuzhen-qualify 2025-spring)
Section: Advanced Questions

Let $k$ be a field of characteristic $p>0$, let $K = k(x,y)$ be the rational function field over $k$ in two variables, and let $F = k(x^p, y^p)$.

1. Prove that $[K:F] = p^2$.

2. Prove that $K^p \subseteq F$.

3. Prove that there are infinitely many intermediate fields of $K/F$.

Problem 8 (qiuzhen-qualify 2025-spring)
Section: Advanced Questions

Let $K$ be the splitting field of $f(x) = x^7 - 7$ over $\mathbb{F}_5$.

1. Determine $\mathrm{Gal}(K/\mathbb{F}_5)$.

2. Find all the intermediate fields of $K/\mathbb{F}_5$.

3. Factorize $f(x)$ into irreducible polynomials.

Problem 9 (qiuzhen-qualify 2025-spring)
Section: Advanced Questions

1. Let $R$ be an integrally closed domain with fraction field $K$. Let $L$ be an algebraic extension of $K$. Show that if $x \in L$ is integral over $R$ then its minimal polynomial over $K$ is in $R[t]$.

2. Let $R = \mathbb{C}[t_1,\ldots,t_n]$ and $f \in R$ be a square-free polynomial. Let $B = R[s]/(s^2 - f)$. Show that $B$ is an integrally closed domain.

Problem 10 (qiuzhen-qualify 2025-spring)
Section: Advanced Questions

Let $R$ be a commutative ring, $I$ be a finitely generated ideal of $R$ such that $I^2 = I$.

1. Show that $I$ is generated by a unique element $e$ such that $e^2 = e$.

2. Suppose I ≠ 0, R0. Show that $R$ is a product of two rings.

3. Find the number of all ideals $J$ of $\mathbb{C}[x]/x(x-1)(x-2)$ such that $J^2 = J$.

Problem 11 (qiuzhen-qualify 2025-spring)
Section: Advanced Questions

Let $S_n$ be the symmetric group of $n$ letters. Construct all the complex irreducible representations for the symmetric groups $S_3$ and $S_4$, and calculate their character tables.

Problem 12 (qiuzhen-qualify 2025-spring)
Section: Advanced Questions

Let $A,B$ be commutative rings.

1. Suppose $B$ is a flat $A$-algebra. Let $a_1,a_2$ be ideals of $A$, show that $a_1B \cap a_2B = (a_1 \cap a_2)B$.

2. Let $k$ be a field, $A = k[x,y]$ and $B = A[z]/(xz - y)$. Show that $B$ is not a flat $A$-algebra.