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I would like to give some details in order to make clear that one can give a proof with hardly any computations at all (I have never looked at the Bourbaki presentation but I guess they make the same …

If the dimension of $V$ is even, $2n$, then there are two families of totally isotropic subspaces of dimension $n$ (if the form is split otherwise there may be no such subspace at all). Two such subsp …

I think all non-archimedean locally compact fields are homeomorphic: Their rings of integers are compact, metric and totally disconnected and hence are all homeomorphic (to the Cantor set). The same i …

Let $R$ be a finite dimensional algebra over $\mathbb Z/2$. Then $\{1\}\neq R^\times$ unless $R=(\mathbb Z/2)^n$. Indeed, if $N$ is the radical of $R$, then $1+N\subseteq R^\times$ so we may assume $R …

We have that $\det T_k$ is a fixed (depending on $n=\dim V$ and $k$ only) power of $\det T$. To see this, as well as getting the power, one can for instance note that $\mathrm{SL}(V)$ is the commutato …

I shall show that the answer is no when $p=2$ (and it seems to me that a somewhat more involved calculation will work for any $p$). We shall show that there exists a vector bundle $\mathcal E$ such th …

Just some comments that are well-known in the theory of toric varieties (and no doubt to other areas as well). What we are asked to determine is membership in a finitely generated submonoid $\Gamma$ o …

The answer is no (and well-known to people working in the representation theory of algebraic groups in positive characteristic). In fact for $V$ finite dimensional and of dimension $>1$ the two vector …