Problem 1 (qiuzhen-qualify 2023-fall)

Assume that $G$ has a finite-index subgroup. Is it true that $G$ always have a finite-index normal subgroup? If it is true, prove it. If it is false, provide a counter example.

Problem 2 (qiuzhen-qualify 2023-fall)

Let $G$ be a finite group. Prove that the following two statements are equivalent.

1. If $g \in G$ and $h \in G$ generate the same cyclic group, then $g$ and $h$ are conjugate in $G$.
2. The characters of complex representations of $G$ are integer-valued.

Problem 3 (qiuzhen-qualify 2023-fall)

Assume ring $R = \mathbb{Z}[\sqrt{-2}]$.

1. Find all the units in $R$ and prove your conclusion.
2. Decide whether $R$ is a UFD or not and prove your conclusion.

Problem 4 (qiuzhen-qualify 2023-fall)

For any positive odd integer $n$, define $\mathbb{C}$-algebra $R_n = \mathbb{C}[x,y]/(x^2 - y^n)$. Let $A_n$ be the normalization of $R_n$, in other words, the integral closure of $R_n$ in its fraction field. Prove that $A_n \cong \mathbb{C}[t]$, and find $\dim_{\mathbb{C}}\, A_n / R_n$.

Problem 5 (qiuzhen-qualify 2023-fall)

Let $R$ be a commutative ring and $m \le n$ be positive integers. Assume $A \in M_{m \times n}(R)$, in other words a $m \times n$ matrix with entries in $R$. Let $I$ be ideal generated by all $m \times m$ minors of $A$. For any $f \in I$, prove that there exists $B \in M_{n \times m}(R)$ such that $AB = f I_m$.

Problem 6 (qiuzhen-qualify 2023-fall)

Let field $K = \mathbb{C}(x)$ and $F = \mathbb{C}(x^3 + x^{-3})$ be a subfield of $K$.

1. Find $[K : F]$.
2. Write down all the intermediate fields $F \subset L \subset K$ in the form of simple extensions $L = F[\alpha]$ over $F$.

Problem 7 (qiuzhen-qualify 2023-fall)

Let $f$ be an irreducible rational polynomial of degree five with exactly three real roots. Is $f$ solvable by radicals or not? Prove your conclusion.

Problem 8 (qiuzhen-qualify 2023-fall)

Let $\mathfrak{sl}_2$ be complex Lie algebra consisting of $2 \times 2$ trace-zero matrices, and $\rho: \mathfrak{sl}_2 \to \mathfrak{gl}_2$ be the standard representation of $\mathfrak{sl}_2$, and let $S_k$ be the $k$-th symmetric product of $\rho$.

1. Prove that $S_k$ is irreducible.
2. Write down the $\mathfrak{sl}_2$ irreducible decomposition of tensor product $S_m \otimes S_l$.

Problem 9 (qiuzhen-qualify 2023-fall)

Let $e_i$ ($i=1,\ldots,4$) be the standard basis for $R^4$. Consider the symmetric bilinear on $R^4$ defined by
$$
(x,y)=x_1y_1+x_2y_2+x_3y_3-x_4y_4.
$$
A basis $f_i$ ($i=1,\ldots,4$) of $R^4$ is called orthonormal if
$$
(f_1,f_1)=(f_2,f_2)=(f_3,f_3)=1,\quad(f_4,f_4)=-1,\quad(f_i,f_j)=0\text{ if }i\ne j.
$$
Suppose $T$ is a Lorentz transformation on $R^4$, that is, $T$ is a linear transformation such that
$$
(Tx,Ty)=(x,y)
$$
for all $x,y\in R^4$. Prove that there exists an orthonormal basis of $R^4$ such that the matrix of $T$ is block diagonal with blocks of the following types:
1. A block of order 1 with entry $\pm1$.
2. A block of order 2 of the form
$$
\begin{pmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{pmatrix}.
$$
3. a block of order 2 of the form
$$
\pm\begin{pmatrix}
\cosh\theta & \sinh\theta \\
\sinh\theta & \cosh\theta
\end{pmatrix}
\quad\text{or}\quad
\pm\begin{pmatrix}
\cosh\theta & \sinh\theta \\
-\sinh\theta & -\cosh\theta
\end{pmatrix}.
$$
4. A block $A$ of order 3 with eigenvalue $\lambda=\pm1$ so that $(A-\lambda I)^3=0$ but $(A-\lambda I)^2\ne0$.