Problem 1 (qiuzhen-qualify 2024-spring)

Let $V$ be a finite dimensional complex vector space. Given a linear transformation $T: V \to V$, prove

1. There exists diagonalizable linear transformation $D: V \to V$ and nilpotent transformation $N: V \to V$, such that $T = D + N$, and $DN = ND$.

2. Such $D$ and $N$ are uniquely determined by $T$.

Problem 2 (qiuzhen-qualify 2024-spring)

Let $A$ and $B$ be positive definite real symmetric matrices of order $n$, prove that $\det(A+B) > \det(A)$.

Problem 3 (qiuzhen-qualify 2024-spring)

Let $G$ be a fixed finite abelian group of order $n$. How many distinct group homomorphisms are there from $G$ to $R/Z$? Prove your conclusion.

Problem 4 (qiuzhen-qualify 2024-spring)

Let $D_{2n}$ be dihedral group of order $4n$. Write down the character table of irreducible complex representations of $D_{2n}$.

Problem 5 (qiuzhen-qualify 2024-spring)

Let $p$ be a prime number and field $F = F_p(x,y)$, $\alpha \in F$. Assume $K$ is the splitting field of $x^p - x - \alpha$ over $F$.

1. Prove that $K/F$ is Galois extension.

2. Find all the possible Galois group $Gal(K/F)$.

Problem 6 (qiuzhen-qualify 2024-spring)

Let $R$ be an integral domain, Aut $R$ the ring automorphism group of $R$. For finite subgroup $G$ of Aut $R$, define

$$R^G = \{ r \in R \mid g(r) = r, \forall g \in G \}.$$

1. Prove that $R$ is integral over $R^G$.

2. Prove that for finite subgroups $G$ and $H$ of Aut $R$, if $R^G = R^H$, then $G = H$.

Problem 7 (qiuzhen-qualify 2024-spring)

Let $k$ be a field, $A$ and $B$ be finitely generated $k$-algebra, and $f: A \to B$ a ring homomorphism such that the restriction of $f$ on $k$ is identity. Prove that for any maximal ideal $m$ of $B$, the inverse image $f^{-1}(m)$ is a maximal ideal of $A$.

Problem unknown (qiuzhen-qualify 2024-spring)

Let $W$ be the real vector space of real trace-zero $2\times 2$ matrices $$W=\{A\in M_{2\times 2}(\mathbb{R})\mid \operatorname{trace}(A)=0\}.$$

1. Find the signature of the symmetric bilinear form defined by $\langle A, A'\rangle = \operatorname{trace}(AA')$.

2. Prove that $P * A = PAP^{-1}$ defines a linear group operation of $\mathrm{SL}(2,\mathbb{R})$ on the space $W$.

3. Use this operation to define a group homomorphism $\varphi\colon \mathrm{SL}(2,\mathbb{R}) \to O_{2,1}$. Here $O_{2,1}$ is the group of linear transformations preserving real symmetric bilinear form of signature $(2,1)$.

4. Find the kernel of this homomorphism.

Problem 9 (qiuzhen-qualify 2024-spring)

Let $p$ be a prime number and $q=p^n$ a power of $p$. Consider the group $\mathrm{GL}(2,\mathbb{F}_q)$ consisting of $2\times 2$ invertible matrices over finite field $\mathbb{F}_q$. Find

1. The number of conjugacy classes in $\mathrm{GL}(2,\mathbb{F}_q)$.

2. The number of elements in each conjugacy class.