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It is optimal for every $d > 0$. One way to make examples is using extension fields. For any $d > 0$, let $k'/k$ be an extension of degree $d$ and consider the norm map $N:k' \rightarrow k$. Choose …

Sit at a table with the books of Borel, Humphreys, and Springer. Bounce around between them: if a proof in one makes no sense, it may be clearer in the other. For example, Springer's book develops ev …

I will give an intrinsic characterization below for what this unit class modulo 12th powers means, which may be viewed as an answer of sorts: it expresses the obstruction to extracting the 12th root …

I don't think the discriminant being square is an issue. This seems best understood by avoiding the "quadratic formula" expression and identifying the cohomological explanation for the appearance of …

Since the case of interest is $W_2(k)$ with perfect $k$ of characteristic $p > 2$, the answer is given by Ioan Berbec's 2009 paper "Group schemes over artinian rings and applications. In that paper ( …

To allow all characteristics, the semisimplicity requirement on the Lie algebras should be replaced with the requirement that the $G_i$ are semisimple as $F$-groups. (In positive characteristic the L …

Some time ago I mentioned a certain open question in an MO answer, and Pete Clark suggesting posting the question on its own. OK, so here it is: First, the setup. Let $X$ be a projective scheme over …

As Hunter and Sean noted, since the inflation map ${\rm{H}}^1(L/F,G(L)) \rightarrow {\rm{H}}^1(F,G)$ is injective and ${\rm{H}}^1(F,G)$ is always finite (Borel-Serre), such an $L$ always exists. Below …

Rob, I am doubtful that in such generality (with $\phi$ presumably meant to act on $E$ by some unspecified endomorphism) there is a reasonable answer. The reason why Jordan canonical form "works" eve …

The correspondence requires the $(\phi,\Gamma)$-module to have the \'etale property for its underlying $\phi$-module, and this plays an essential role in the proof of the correspondence (see Fontaine' …

It is a standard and important fact that any smooth affine group scheme $G$ over a field $k$ is a closed $k$-subgroup of ${\rm{GL}}_n$ for some $n > 0$. (Smoothness can be relaxed to finite type, but …

Let $k$ be a field complete with respect to a non-archimedean absolute value, and $X$ a separated algebraic space of finite type over $k$. If $X$ is a scheme then $X(k)$ inherits a natural (Hausdor …

Johnson, you have one of the foremost experts in the world on such matters (over general fields) just upstairs from your office. Make use of that. Many of the basic constructions work for split gr …

I assume $G$ is affine. The quick answer is that in the simply connected case it says $1 = 1/1$ by various hard ingredients, and then it is a kind of (not easy) game with Galois cohomology and structu …

I agree with Adam Smith that the question seems a bit misguided, but let me show anyway that the answer is negative away from certain silly cases. Well, first to make a more well-posed question, one …