# Tag Info

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### Motivating Lubin-Tate theory

Sorry I didn’t see this earlier. My memory is vague, and probably colored by subsequent events and results, but here’s how I recall things happening. Since I had read and enjoyed Lazard’s paper on ...
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### What is a tamely-ramified Weil-Deligne representation?

$\def\R{\mathbf{R}}$ $\def\Z{\mathbf{Z}}$ $\def\Q{\mathbf{Q}}$ $\def\Qbar{\overline{\Q}}$ $\def\F{\mathbf{F}}$ $\def\GL{\mathrm{GL}}$ $\def\Gal{\mathrm{Gal}}$ Here are some thoughts on your question ...

### Does the discriminant of an irreducible polynomial of a fixed degree determine the discriminant of the number field it generates?

The discriminants of the irreducible polynomials $$(x^2-2)^2+60 = x^4 - 4 x^2 + 64, \qquad (x^2+2)^2+60 = x^4 + 4 x^2 + 64$$ are both equal to $58982400 = 2^{18} \cdot 3^2 \cdot 5^2$. However, the ...
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Let us start with (b) => (a). We only need to show that if $\mathcal{G}$ is a reductive $O$-group scheme, then $\mathcal{G}_F$ is quasi-split and split after an unramified extension. Let us first ...
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### Is there any explicit description of the maximal totally ramified extension of $\mathbb{Q}_p$?

A composite of totally ramified extensions need not be totally ramified: Example 1. (As per LSpice's suggestion) Consider the extensions $\mathbb Q_p(\sqrt{p})$ and $\mathbb Q_p(\sqrt{\varepsilon p})$...

### Are the abelian absolute Galois groups of these local fields isomorphic?

This is how to answer the question (but it's not an answer). (Edit: it and the comments below now form an answer). Let $K_1$ and $K_2$ be the fields in question. By local class field theory you're ...
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### Is $G \rightarrow G/P$ surjective on $K$-points over a local field?

The map G(K) to (G/P)(K) is surjective over any field K. Here is a link to an explanation by Brian Conrad. http://math.stanford.edu/~conrad/249CS13Page/handouts/parsurj.pdf

### Which groups are Galois over some p-adic field?

When $p \neq \ell$, if $N/K$ has Galois group $G$ then $N/K$ is tamely ramified. It follows that $N = K(\sqrt[e]{\pi}, \zeta)$ where $e$ is the ramification degree of $N/K$, $\pi$ is some uniformizer ...

### Are the abelian absolute Galois groups of these local fields isomorphic?

Although @znt has gotten the answer through pari, I think it may be instructive to outline my argument. It all depends on the transition function of Higher Ramification Theory: for a finite extension ...

### Reference request for Kato's paper: A generalization of local class field theory by using K -groups

I found this old question while searching for Kato's paper myself. Just in case anyone else is also still looking for these, here's what I found. Kato's work was published in three installments in J. ...
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### Is the intersection of ramification groups in upper numbering of a $p$-adic local field trivial?

Yes, at least if the upper ramification groups $G^\nu$ are defined as $\varprojlim_L\mathrm{Gal}(L/K)^\nu$ for $L/K$ finite Galois (e.g. as in ). This makes sense because the upper-numbering is ...
In the "more sophisticated" direction, we can ask a similar question about the moduli stack $\mathscr{M}_g$ of hyperelliptic curves of genus $g$. If $K$ is a topological field, there is a ...