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Let $k$ be a $p$-adic field and $G$ be a connected non-abelian reductive algebraic group over $k$. I am asking for a proof of the following lemma: Lemma. Assuming that $p$ is "good" for $G$, there ex …

Here I give details of the reduction in LSpice's comment. I write it as an answer rather than a string of comments in order to have an editable text. The reduction goes as follows. According to Will's …

Let $T\subset {\rm GL}(N,{\mathbb Q})$ be an $n$-dimensional ${\mathbb Q}$-torus given by its Lie algebra $\mathfrak{t}\subset \mathfrak{gl}(N,{\mathbb Q})$. I want to compute, in some sense explicit …

There is quite a lot of work on computing Galois groups of splitting fields of polynomials over $\Bbb Q$. Magma is quite good at it. In this answer to Decomposition groups for the Galois module $\mu_8 …

In addition to the answer of Gro-Tsen: Let $G=\operatorname{SU}(3)$. Then $H^1({\mathbb R},G)$ classifies matrices of Hermitian forms in 3 variables with determinant 1. There are two equivalence class …

$\require{AMScd} $In this answer, $k$ is a nonarchimedean local field. Lemma 1. Consider a short exact sequence of linear algebraic $k$-groups $$ 1\to C\overset i\longrightarrow G\overset j\longright …

$\newcommand{\diag}{{\rm diag}} \newcommand{\sH}{{\mathcal H}} \newcommand{\R}{{\mathbb R}} \newcommand{\HH}{\sf H} \newcommand{\V}{{\sf V}} \newcommand{\B}{{\sf B}} \newcommand{\C}{{\Bbb C}} $No, th …

Let $K$ be an arbitrary field, and let $K^s$ be a fixed separable closure of $K$. Let $F/K$ a be a finite Galois extension in $K^s$. Let $n>0$ be a natural number. Let $A$ be a central simple algebra …

Let $G$ be a finite group, and let $M$ be a finitely generated $G$-module, that is, a finitely generated abelian group on which $G$ acts. We work with the first homology group $$ H_1(G,M).$$ For any c …

$\newcommand{\ab}{{\rm ab}} \newcommand{\ord}{{\rm ord}} $Let $G$ be a finite or profinite group. Consider the abelianized group $$G^\ab=G/G'$$ where $G'$ is the commutator subgroup of $G$. Let $H\sub …

$\newcommand{\nN}{{\mathcal N}} \newcommand{\SL}{{\rm SL}} \newcommand{\G}{{\bf G}} \newcommand{\Q}{{\Bbb Q}} \newcommand{\R}{{\Bbb R}} \newcommand{\C}{{\Bbb C}} $The answer is No. Write $\nN$ for the …

This seemingly elementary question was asked in Mathematics StackExchange.com: https://math.stackexchange.com/q/4779592/37763. It got upvotes, but no answers or comments, and so I ask it here. Let $G$ …

$\newcommand{\coker}{{\rm coker}} $ I construct the required homomorphism. Let $A$ be an abelian variety over a number field $K$, and let $L/K$ be a finite Galois extension. We denote by $A'$ the dual …

$\newcommand{\wt}{\widetilde}$ Let $n=1,2$. There are infinite torsion abelian groups $H^1$, $H^2$ killed by some natural number $m$. There are finite subgroups $$ {\rm Sha}^1 \subset H^1,\quad …

Let $F$ be a global function field, for example $F={\mathbb F}_q(t)$, the field of rational functions in one variable over a finite field ${\mathbb F}_q\,$. Question. What would be an example of a gl …