Questions tagged [local-field]
For questions about local field, which is a special type of field that is a locally compact topological field with respect to a non-discrete topology.
551 questions
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Local to global and Hensel's Lemma
I have a silly question. I am clearly missing a subtlety with the use of Hensel's Lemma.
Suppose I have a variety $S/\mathbb{Q}$. I wish to determine all of its $\mathbb{Q}_2$ points. I search $\mod 4$...
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Original version of Bourbaki's _Algebra_?
I'm reading Serre's Local Fields, where he quoted a lot of Bourbaki's Algèbre. These citations don't line up with the modern version of the book (which I got from Springer's website), but seem to ...
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Compact Zariski dense subgroup of rational points of semisimple algebraic group over local fields
Let $\mathbb{Q}_{p}$ be the field of $p$-adic numbers and let $\mathcal{G}$ be a semisimple algebraic group over $\mathbb{Q}_{p}$. Let $\Gamma$ be a topologically finitely generated, compact subgroup ...
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Show that $S^{-1}B/S^{-1}I$ and $B/I$ are isomorphic
Im trying to understand a step in a proof by Serre in his book "local fields", chapter 1, proposition 10.
Let $L/K$ be a local field extension. Let $A$ be the ring of integers of $K$ and let ...
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Let $r_n$ be the least $N$ s.t. every $p^n$-th power mod $\mathfrak{m}_K^N$ lifts. Do we almost have $r_n=r_{n-1}+e$ for $e$ the ramification index?
$\renewcommand{\O}{\mathcal{O}_K}
\newcommand{\m}{\mathfrak{m}_K}$
I'm afraid that this post is much too long, but I do want to present my motivations and ideas below. I would be really sorry if you ...
- 2,877
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Additive discrete group actions of self-Pontrjagin dual topological rings with pro-Lie stabilizer
Let's talk about Pontrjagin self-dual commutative topological rings. To define a special condition, fix a topological ring $A$. Suppose that:
There is an additive topological subgroup $A_0$ of $A$ ...
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Two questions on Deuring's Lifting Theorem
I have two questions about Deuring's Lifting Theorem:
Let $E/\mathbb F_q$ be an elliptic curve over a finite field and let $\phi \in End(E)$ be nonzero. There exists an elliptic curve $E^*$ over a ...
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$N_{L/K}(U_L^i)\supseteq U_K^i$
I need to prove that $N_{L/K}(U_L^i)\supseteq U_K^i$, where $K$ is a $p$-adic number field, $L/K$ is a finite unramified field extension, and $$U_L^i=\begin{cases}\mathcal O_L^\times, &i=0,\\
1+\...
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Is there a canonical choice of algebraic closures for finite fields?
From what I understand, there is an almost canonical choice for an algebraic closure of $\mathbb{Q}$. Namely, take all algebraic elements in $\mathbb{C}$. As the extension $\mathbb{R}/\mathbb{Q}$ is ...
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Choosing a desired embedding of $\overline{\mathbb{Q}}$ in $\overline{\mathbb{Q}}_p$
Suppose I have a local field $L/\mathbb{Q}_p$ with prime $\mathfrak{p}$. I want to take an embedding $\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_p$ so that I can see $G(\overline{\...
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Do we have $[\sqrt{d}]\pi_E+\pi_E[\sqrt{d}]=0$ for the supersingular reduction of CM elliptic curves?
Let $p$ be a prime, $d<0$ a discriminant, $K/\mathbb{Q}_p$ Galois, $A_K\subset K$ the valuation ring, $\mathfrak{m}_K\subset A_K$ the maximal ideal, $k:=A_K/\mathfrak{m}_K$ the residue field. Let $...
- 9,184
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Unramified iff norm map is surjective on the units?
Let $L/K$ be a finite extension of local fields (or maybe complete discrete valued fields), not necessarily abelian. Is it true that $L/K$ is unramified if and only if the norm map $\mathcal{O}_L^*\to ...
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Is $\operatorname{PGL}_n(k)$ unimodular, where $k$ is $\mathbb R$, $\mathbb C$, or $\mathbb Q_p$?
I know that $\operatorname{GL}_n(k)$ is unimodular. I would really be happy if it was the case for $\operatorname{PGL}_n$.
I don't know anything about algebraic groups, but I think we could use the ...
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"Unramified" extensions of a $p$-adic field with prescribed algebraically closed residue fields
Let $E$ be a finite extension of $\mathbb Q_p$, with residue field $\kappa_E = \mathbb F_q$ for some power $q$ of the prime number $p$. One can consider $\breve E$, the completion of the maximal ...
- 6,422
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Weil height of algebraic number
We know the basic fact that if $n \equiv 0 \pmod p$ then $|n| \geq p$ (provided $n \neq 0$). Let $\alpha$ be a non-zero algebraic number and suppose that there is a prime ideal $\mathcal{P}$ in $\...
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