Problem 1 (yau-contest 2023)
Section: Algebra and Number Theory

Let $F$ be a field. Let $n \ge 2$ be an integer. For $1 \le i \ne j \le n$ and for $\lambda \in F$, set $E_{ij}(\lambda) = (e_{kl})$ the following $n \times n$ matrix such that

$$
 e_{kl}=\begin{cases}
 1, & \text{if } k=l;\\
 \lambda, & \text{if } (k,l)=(i,j);\\
 0, & \text{otherwise.}
 \end{cases}
$$

(1) Show that the special linear group $\mathrm{SL}_n(F)$ can be generated by the matrices $E_{ij}(\lambda)$ ($1 \le i \ne j \le n$ and $\lambda \in F$).

(2) Assume $n \ge 3$. Show that the general linear group $\mathrm{GL}_n(F)$ is not solvable. Hint: one can start by computing the commutators of the matrices $E_{ij}(\lambda)$.

(3) Assume $n=2$ and $F$ is a field containing at least $4$ elements. Show that $\mathrm{GL}_n(F)$ is not solvable either. Hint: one can start by computing the commutator of the matrix $E_{12}(\lambda)$ with a diagonal matrix.

(4) Let $\mathbb{F}_q$ denote the finite field with $q$ elements. Are $\mathrm{GL}_2(\mathbb{F}_2)$ and $\mathrm{GL}_2(\mathbb{F}_3)$ solvable? Justify your assertion.

Problem 2 (yau-contest 2023)
Section: Algebra and Number Theory

Let $R$ be a ring with identity $1 \ne 0$.

(1) Assume that $R$ is commutative. Show that, if we have an isomorphism $R^n \simeq R^m$ of $R$-modules for some positive integers $n$ and $m$, then $m=n$.

(2) The analogous assertion of (1) for a non-commutative ring is not true in general as shown by the following example (in particular, the notion of "rank" of a finite free module over such rings is not well-defined). Let $K$ be a non-trivial (not necessarily commutative) ring with identity, and $F$ be a free $K$-module with a countable basis indexed by $\mathbb{Z}_{\ge 1}$:

$$e_1,\ldots,e_2,\ldots,e_n,\ldots.$$

Write $R=\mathrm{End}_K(F)$, which is a ring with the usual addition and composition of two $K$-linear endomorphisms of $F$.

(a) Show that $R$ is not commutative.

(b) Let $f_0 \in R$ (resp. $f_1 \in R$) such that $f_0(e_{2i})=e_i$ and $f_0(e_{2i-1})=0$ (resp. $f_1(e_{2i})=0$ and $f_1(e_{2i-1})=e_i$) for every $i \in \mathbb{Z}_{\ge 1}$. Show that $\{f_0,f_1\}$ is a basis of $R$ as a left $R$-module (given by left-multiplication).

(c) For any integers $n,m \ge 1$, show that there exists an isomorphism $R^n \simeq R^m$ of $R$-modules.

Problem 3 (yau-contest 2023)
Section: Algebra and Number Theory

Let $G$ be a profinite group. We say that it is topologically finitely generated if there exist finitely many elements $g_1,\ldots,g_n$, such that the closed subgroup generated by $g_1,\ldots,g_n$ is equal to $G$.

(1) Assume that $G$ is topologically finitely generated. Show that, for a fixed positive integer $n$, $G$ has only finitely many open subgroups of index $n$.

(2) Show that, for $K$ a number field (i.e., a finite field extension of $\mathbb{Q}$), its absolute Galois group $G_K$ is not topologically finitely generated.

Problem 4 (yau-contest 2023)
Section: Algebra and Number Theory

Let $R$ be a commutative ring with identity. An $R$-module $M$ is said to be of finite presentation if there exists an exact sequence of $R$-modules

$$R^n \longrightarrow R^m \longrightarrow M \longrightarrow 0$$

for some positive integers $m,n \ge 0$.

(1) Let $f:N \to M$ be a surjective morphism of $R$-modules with $N$ of finite type and $M$ of finite presentation. Show that the $R$-module $\ker(f)$ is of finite type.

(2) Let $M$ be an $R$-module of finite type. Show that the following statements are equivalent.

(a) $M$ is flat and of finite presentation.

(b) $M$ is locally free, i.e., there exist $f_1,\ldots,f_s \in R$ generating the unit ideal of $R$ such that $M_{f_i}$ is free over $R_{f_i}$.

(c) $M$ is projective as an $R$-module.

(3) Let $S=\prod_I R$ the product of countably many copies of $R$, and $I=\bigoplus_I R \subset S$. Show that $S/I$ is a flat $S$-module which is not projective.

Problem 5 (yau-contest 2023)
Section: Algebra and Number Theory

Let $p$ be a prime number. Let $\mathbb{Z}_p$ be the ring of $p$-adic integers.

(1) Let $a \in \mathbb{C}$ such that $\lim_{r \to \infty} a^{p^r} = 1$. Show that $a$ is a $p^m$-th root of unity for some $m$.

(2) Let $A \in M_{n\times n}(\mathbb{C})$ be an $n \times n$ complex matrix such that $\lim_{r \to \infty} A^{p^r} = I_n$, with $I_n$ the identity matrix. Show that $A^{p^m} = I_n$ for some integer $m$.

(3) Determine, up to isomorphisms, all the finite-dimensional continuous complex representations of $\mathbb{Z}_p$.

Problem 6 (yau-contest 2023)
Section: Algebra and Number Theory

Let $p$ be a prime number. Let $K$ be a $p$-adic local field, i.e., a finite field extension of $\mathbb{Q}_p$. Denote by $\mathcal{O}_K$ its ring of integers, with $\pi \in \mathcal{O}_K$ a uniformizer.

(1) Show that, the following logarithm map

$$\log : 1 + p\mathcal{O}_K \longrightarrow p\mathcal{O}_K, \quad 1+x \mapsto x - \frac{x^2}{2} + \frac{x^3}{3} + \cdots$$

is a well-defined isomorphism of groups, which can be extended in a unique way to a group homomorphism

$$\log_{\pi} : K^{\times} \longrightarrow K,$$

such that $\log_{\pi}(\pi)=0$. Determine the kernel of $\log_{\pi}$.

(2) With the help of the logarithm above, show that for any integer $n \ge 1$, the group quotient $K^{\times}/K^{\times,n}$ is a finite group.