Problem 1 (yau-contest 2024)
Section: Algebra and Number Theory

Let $G$ be a finite group.

(1) Let $K$ be a field. Show that $G$ has a finite-dimensional faithful $K$-linear representation.

(2) Show that $G$ has a faithful one-dimensional complex representation if and only if $G$ is cyclic.

(3) Assume moreover that $G$ is commutative. Let $n \ge 1$ be an integer. Show that $G$ has a faithful $n$-dimensional complex representation if and only if $G$ can be generated by $n$ elements.

(4) Classify all finite groups having a faithful $2$-dimensional real representation.

Problem 2 (yau-contest 2024)
Section: Algebra and Number Theory

Let $n \ge 1$ be an integer. Let $A$ be a discrete valuation ring with $K$ its field of fractions and $\pi \in A$ a uniformizer. For $\lambda=(\lambda_1,\ldots,\lambda_n)\in \mathbb{Z}^n$ write
\[
D_\lambda=\operatorname{diag}(\pi^{\lambda_1},\ldots,\pi^{\lambda_n})=\begin{pmatrix}
\pi^{\lambda_1}&0&\cdots&0\\
0&\pi^{\lambda_2}&\cdots&0\\
\vdots&\vdots&\ddots&\vdots\\
0&0&\cdots&\pi^{\lambda_n}
\end{pmatrix}\in \mathrm{GL}_n(K).
\]
Show that, for $\lambda,\mu\in\mathbb{Z}^n$, the following intersection inside $\mathrm{GL}_n(K)$
\[
\mathrm{GL}_n(A)\cdot D_\mu\cdot \mathrm{GL}_n(A)\;\cap\; U(K)\cdot D_\lambda
\]
is non-empty if and only if $\lambda_{\mathrm{dom}}\le \mu_{\mathrm{dom}}$. Here

- $\mathrm{GL}_n(K)$ (resp. $\mathrm{GL}_n(A)$) is the group of invertible $n\times n$ square matrices with coefficients in $K$ (resp. in $A$), and $U(K)\subset \mathrm{GL}_n(K)$ is the standard unipotent subgroup, that is, the subgroup of upper triangular matrices with coefficients $1$ on the diagonal.
- For $\alpha=(a_1,\ldots,a_n)$ and $\beta=(b_1,\ldots,b_n)$ two elements in $\mathbb{Z}^n$, we write $\alpha\le\beta$ if
\[\sum_{i=1}^k a_i\le \sum_{i=1}^k b_i,\quad\text{for any }1\le k\le n,\]
and if $\sum_{i=1}^n a_i=\sum_{i=1}^n b_i$. Write also $\alpha_{\mathrm{dom}}:=(a'_1,\ldots,a'_n)$ with $a'_1,\ldots,a'_n$ an arrangement of $a_1,\ldots,a_n$ such that $a'_1\ge a'_2\ge\cdots\ge a'_n$.

Problem 3 (yau-contest 2024)
Section: Algebra and Number Theory

Let $k$ be an imperfect field of characteristic $p>0$. Let $a\in k\setminus k^p$.

(1) Show that the polynomial $X^p-a\in k[X]$ is irreducible.

(2) Let $A=k[X]/(X^{p^2}-aX^p)$. Compute $A_{\mathrm{red}}$, the quotient of $A$ by its nilpotent radical.

Problem 4 (yau-contest 2024)
Section: Algebra and Number Theory

Let $k$ be a field of character $p>0$ and consider $k(t)\subset k((t))$.

(1) Show that the field extension $k((t))/k(t)$ is transcendental, i.e., $k((t))$ contains at least one transcendental element over $k(t)$. (Hint: don’t spend too much time on this question.)

(2) Let $\alpha\in k[[t]]$ which is transcendental over $k(t)$, and write $\beta=\alpha^p\in k[[t]]$. Let $K=k(t,\beta)$ and $L=k(t,\alpha)$: both are subfields of $k((t))$. Let $A=k[[t]]\cap K$. Determine the integral closure $B$ of $A$ in $L$, and show that $A$ and $B$ are two discrete valuation rings.

(3) Is $B$ finitely generated over $A$ as a module? Justify your assertion.

Problem 5 (yau-contest 2024)
Section: Algebra and Number Theory

Let $p$ be a prime number. Consider $\mathbb{Z}_p$ (resp. $\mathbb{Q}_p$) the ring of $p$-adic integers (resp. field of $p$-adic numbers). Clearly $\mathbb{Z}\subset \mathbb{Z}_p$.

(1) Show that the set $\mathbb{Z}$ is dense in $\mathbb{Z}_p$, and deduce that a map $f:\mathbb{Z}\to\mathbb{Q}_p$ can be extended to a continuous function on $\mathbb{Z}_p$ if and only if $f$ is uniformly continuous, i.e., for any $\varepsilon>0$, there exists some integer $N>0$ so that $|f(n)-f(m)|<\varepsilon$ for any integers $n,m\in\mathbb{Z}$ with $m\equiv n\pmod{p^N}$.

(2) Let $a\in\mathbb{Q}_p\setminus\{0\}$. Under what condition on $a$, the map
\[f:\mathbb{Z}\longrightarrow\mathbb{Q}_p,\qquad n\mapsto a^n\]
can be extended to a continuous function over $\mathbb{Z}_p$? Justify your assertion.

(3) Assume that the condition in (2) is fulfilled. Can we even extend the function $f$ in (2) to a continuous map
\[a^x:\mathbb{Q}_p\longrightarrow\mathbb{Q}_p,\]
so that $a^{x+y}=a^x a^y$ for any $x,y\in\mathbb{Q}_p$? Justify your assertion.

Problem 6 (yau-contest 2024)
Section: Algebra and Number Theory

Let $n\ge 1$ be an integer and write $\Phi_n(X)$ the $n$-th cyclotomic polynomial, that is, the minimal polynomial of a primitive $n$-th root of unity in $\mathbb{C}$ over $\mathbb{Q}$. Write also $\varphi(n)=\deg(\Phi_n(X))$.

(1) Let $q$ be a power of a prime number such that $(q,n)=1$. Show that $\Phi_n$, viewed as an element in $\mathbb{F}_q[X]$, can be decomposed as a product of $\varphi(n)/d$ irreducible polynomials of degree $d$, with $d$ the order of $q$ in the multiplicative group $(\mathbb{Z}/n\mathbb{Z})^\times$.

(2) From now on, assume $n=2^{r+1}$ for some integer $r\ge 1$. Let $\zeta=\zeta_n$ be a primitive $n$-th root of unity and $K=\mathbb{Q}[\zeta]$. Let $p$ be a prime with $p\equiv -3\pmod{8}$.

(a) For $x,y\in K=\mathbb{Q}[\zeta]$, define
\[(x,y):=\sum_{\tau} \tau(x)\cdot \overline{\tau(y)}\]
where $\tau$ runs through all the embeddings $K\hookrightarrow \mathbb{C}$ of $K$ into the field $\mathbb{C}$ of complex numbers. Write $K_{\mathbb{R}}=K\otimes_{\mathbb{Q}}\mathbb{R}$, and we use the same notation to denote the (a priori $\mathbb{C}$-valued) bilinear form on $K_{\mathbb{R}}$ obtained by extension of scalars. Show that $(\cdot,\cdot)$ gives an inner product on $K$ and for $0\le i,j<2^r$,
\[
(\zeta^i,\zeta^j)=\begin{cases}
2^r,& \text{if } i=j,\\
0,& \text{otherwise.}
\end{cases}
\]
In particular, we obtain an Euclidean space $K_{\mathbb{R}}$ and $(\zeta^i/\sqrt{2^r})_{0\le i<2^r}$ is an orthonormal basis.

(b) Decompose $p\,\mathcal{O}_K$ into a product of prime ideals.

(c) Let $\mathfrak{p}\subset\mathcal{O}_K$ be a prime ideal of $\mathcal{O}_K$ containing $p$. Show that for every $\alpha\in\mathfrak{p}$, $|\alpha|^2\in 2^r p\,\mathbb{Z}$, and compute the length of the shortest non-zero vector in the prime ideal $\mathfrak{p}\subset K_{\mathbb{R}}$.