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Let $k$ be a field of characteristic 0, and $\overline k$ be a fixed algebraic closure of $k$. Let $k\subset F\subset E$ be a tower of finite Galois extensions in $\overline k$, where both $\mathrm{Ga …

I translate into English Lemma 6.5 from Sansuc's paper Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. reine angew. Math. 327 (1981), 12-80. Let $X$ be …

Let $k$ be an arbitrary field, $d\in k$, and $d$ is not a square in $k$. Consider the binary quadratic form $f(x,y)=x^2-d y^2$ (it is the norm from $k(\sqrt{d})$ to $k$). I am looking for a reference …

Consider the field of fractions $K$ of the quotient algebra $\mathbb{R}[x,y,z,t]/(x^2+y^2+z^2+t^2+1)$, where $\mathbb{R}$ is the field of real numbers and $x,y,z,t$ are variables. Clearly $-1$ is a su …

This is a simplified version of the accepted answer of R. van Dobben de Bruyn. I use his notation. It suffices to prove the second assertion of the question; see the comment of R. van Dobben de Bruyn …

Let $k_0$ be a field of characteristic 0, $k/k_0$ be a quadratic extension, and $A/k$ be a central simple algebra over $k$ of dimension $9=3^2$ with an involution of second kind $\sigma$. Then $^\sigm …

Let $G$ be a finite group. By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$. One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis permut …

$\newcommand{\Tors}{{\rm Tors}} \newcommand{\tf}{{\rm\, t.f.}} \newcommand{\Gt}{{\Gamma\!,\,\Tors}} \newcommand{\Gtf}{{\Gamma\!,\tf}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\Z}{{\mathbb Z}} \newcomm …

$ \newcommand{\G}{\Gamma} \newcommand{\rsa}{\rightsquigarrow} \newcommand{\Z}{{\mathbb Z}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\Lam}{\Lambda} \newcommand{\Tor}{{\rm Tor}} \newcommand{\Gt}{{\Gamma …

Let $k_0$ be a field of characteristic 0, and let $k$ is a fixed algebraic closure of $k_0$. Write $G={\rm Gal}(k/k_0)$. Let $A_0$ be a finitely generated commutative $k_0$-algebra with a unit. Then …