Problem 1 (yau-contest 2019)
Section: Algebra and Number Theory

Let $S_n$ be the group of permutations of $\{1,2,\ldots,n\}$. Let $\sigma \in S_n$ be the permutation
\[(1,n)(2,n-1)\cdots (k,n-k+1)\cdots .\]
Prove that the centraliser $Z_{S_n}(\sigma)$ is isomorphic to $S_{[i]} \wr (\mathbb{Z}/2\mathbb{Z})^{[i]}$.

Problem 2 (yau-contest 2019)
Section: Algebra and Number Theory

Recall that the algebra of regular functions on a vector space $W$ is the symmetric algebra of linear forms on $W$.

Let $V=\mathbb{C}^2$. Let $\mathbb{C}[\operatorname{End}_{\mathbb{C}}(V)]$ be the algebra of regular functions on $\operatorname{End}_{\mathbb{C}}(V)$. The natural action of the group $G=\mathrm{SL}_2(\mathbb{C})$ on $V$ induces an action of $G\times G$ on $\operatorname{End}_{\mathbb{C}}(V)$ by left and right multiplication. Thus we get an action of $G\times G$ on $\mathbb{C}[\operatorname{End}_{\mathbb{C}}(V)]$.

Compute the algebra of fixed points $\mathbb{C}[\operatorname{End}_{\mathbb{C}}(V)]^{G\times G}$.

Problem 3 (yau-contest 2019)
Section: Algebra and Number Theory

Let $R$ be a Noetherian ring and $I \subset R$ be an ideal. Define the Rees algebra as
\[\operatorname{Rees}(I,R):=\bigoplus_{n\ge 1} I^n t^n \subset R[t].\]
Prove that $\operatorname{Rees}(I,R)$ is Noetherian.

Problem 4 (yau-contest 2019)
Section: Algebra and Number Theory

Let $p$ be an odd prime. Let $\Phi_p$ be the $p$-th cyclotomic field, i.e., $\Phi_p=\mathbb{Q}(\zeta_p)$ where $\zeta_p$ is a primitive $p$-th root of unity.

1. Show that $\Phi_p/\mathbb{Q}$ is a Galois extension with Galois group $(\mathbb{Z}/p\mathbb{Z})^{\times}$.
2. Deduce that $\Phi_p$ contains a unique quadratic extension of $\mathbb{Q}$.
3. Write $g_p$ for the Gauss sum $\sum_{a=1}^{p-1} \left(\frac{a}{p}\right)\zeta_p^a$. Show that
   - $\overline{g_p}=\left(\frac{-1}{p}\right)g_p$,
   - $|g_p|^2=p$.
4. Determine the unique quadratic extension of $\mathbb{Q}$ contained in $\Phi_p$.

Problem 5 (yau-contest 2019)
Section: Algebra and Number Theory

1. Let $E/F$ be a finite Galois extension. Assume that the Galois group $\operatorname{Gal}(E/F)$ is generated by a single element $\sigma$. Let $x\in E$ be such that $\operatorname{tr}_{E/F}(x)=0$. Show that there exists $y\in E$ such that $x=\sigma(y)-y$.

2. Let $F$ be a field of characteristic $p$, and let $E/F$ be a Galois extension of degree $p$. Show that there exists $x\in E$ such that $E\cong F[T]/(T^p-T-x)$.