The local-fields tag has no wiki summary.

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### Measure of ramification of local fields using upper numbering

We let $F$ be a non-archimedean local field (say with finite residue field). Consider a Galois extensions $E$ of $F$, with $G = Gal(E/F)$, in a fixed separable closure $\bar{F}$ of $F$. Considering ...

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### Can we prove the uniqueness of the local Artin map by using mostly global class field theory?

Let $l/k$ be a finite abelian extension of $p$-adic fields. There is a well defined local Artin map $k^{\ast} \rightarrow Gal(l/k)$ with kernel $N_{l/k}(l^{\ast})$. Let's suppose that we have only ...

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### Swan conductor, representation Weil group

Let $F$ be a non-archimedean local field and $\mathcal W_F$ its Weil group. We consider a linear representation $\sigma$ of $\mathcal W_F$. Could someone explain to me the definition of the Swan ...

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### $K^{ur}K^{\pi} = L$

Let $K$ be a $p$-adic field, and $L$ an infinite abelian extension of $K$ containing $K^{ur}$. Let $\Phi: K^{\ast} \rightarrow Gal(L/K)$ be the local Artin map. Let $\pi$ be a uniformizer for $K$, ...

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### Is $K^{ur} K^{\pi} = L$?

Let $L/K$ be a finite extension of $p$-adic fields, $\pi$ a uniformizer of $K$, $\theta = (-, L/K)$ the local Artin map $K^{\ast} \rightarrow Gal(L/K)$, $E$ be maximal unramified extension of $K$ ...

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### split tori over local fields

Let $F$ be a non-archimedean local field, and $\mathscr O$ its ring of integers. Suppose $T$ is an $F$-split torus, i.e., $T = (\mathbb G_m)^r$ where $\mathbb G_m$ denotes the multiplicative group. ...

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### Weil group of a local field, small notational problem

In Bushnell and Henniart, The Local Langlands conjecture for GL(2), there is a proposition on p. 184 in which they prove the following:
Let $F$ be a non-archimedean local field, $\mathcal W_F$ its ...

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### Is every connected reductive group over a local field already defined over a global field?

Let $K$ be a local field, e.g. $\mathbb{Q}_p$ or $\mathbb{F}_p((t))$. Let $G$ be a connected reductive group over $K$. Is it true that $G$ is already defined over a global field? More precisely, does ...

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### Structure of locally compact non discrete topological division algebras without the use of Haar measure

There is a well-known structure theorem for locally compact non discrete topological division algebras, see here
http://math.stackexchange.com/q/1160086/187521
(I repost it here because I think it ...

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### Automorphisms of k((X))

I'm looking for a good reference for the following fact:
Let $k$ be a perfect field of characteristic $p$ and let $K=k((X))$.
Then every $k$-linear automorphism of $K$ is continuous with respect
to ...

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### A good book on adeles and ideles

Many results in number theory are stated either in a classical language or in an adelic one. I am often impressed of the efficiency and the satisfactory computational properties of the adelic setting, ...

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### Is $G \rightarrow G/P$ surjective on $K$-points over a local field?

Let $K$ be a local field, $G$ a (connected) reductive $K$-group, and $P \le G$ a parabolic subgroup. Is the map $G(K) \rightarrow (G/P)(K)$ necessarily surjective, and, if so, then why?

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### On unramified p-adic groups

Let G be a reductive group over a local field F. Let O be the ring of integers of F.
The following are equivalent (and groups satisfying these conditions are called unramified):
(a) G is quasisplit ...

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### Compact elements in $G(K)$ for a reductive group $G$ over a nonarchimedean local field $K$

Let $K$ be a nonarchimedean local field and $G$ a (connected) reductive group over $K$, so that $G(K)$ carries a natural topology. An element $g \in G(K)$ is compact if it is contained in a compact ...

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### Relation between 1-dimensional and 2-dimensional reciprocity maps

Let $M/L/\mathbb{Q}_p$ be a finite galois abelian extension of local fields and define
$\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathcal{Z}}a_iT^i : a_i\in M, \min_{i\in \mathcal{Z}} v(a_i)>-\infty, ...

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### Theorem 7b of Serre's “Propriétés galoisiennes des points d'ordre fini des courbes elliptiques”

Could someone please point me towards a proof of the statement in the second paragraph, in the proof of Theorem 7b of Serre's Propriétés galoisiennes...? The statement is as follows:
Let $F$ and $F'$ ...

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### explicit uniformizer for the false Tate extension

Let $p$ be an odd prime and let $n\geq 1$. Set $K=\mathbb{Q}_p(\zeta_{p^n})$,
$L=\mathbb{Q}_p(\sqrt[p^n]{p})$, and $M=KL$. I claim that $M$ is totally ramified of degree $\phi(p^n)p^n$ (the proof ...

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### Finite extension of local fields

Can a (higher) local field have uncountably many finite (seperable) extensions?

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### Compact subgroups of linear groups over nonarchimedean fields

Let $n \in \mathbb{N}$, $K$ a (nonarchimedean) local field, $\overline{K}$ its algebraic closure. Take a compact subgroup $G \leq \text{GL}_n(\overline{K})$. Must there be a finite extension $F$ of ...

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### Continuity of the solutions of an isogeny in a formal group

Notation for the problem:
Let $E/\mathbb{Q}_P$ be a local field, and $\mu_E$ its maximal ideal. Let $K=E\{\{T\}\}$ be the standard 2-dimensional local field equipped with the Parshin topology and let ...

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### Why is the norm map dual to restriction under Tate local duality?

Let $L/K$ be a finite Galois extension of nonarchimedean local fields, and let $A$ and $A^t$ be dual abelian varieties over $K$. Tate local duality tells us that $A^t(K)$ and $H^1(K, A)$ are ...

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### Parshin's buildings for higher local fields

What is the status of the theory of buildings for higher local fields?
I know that there are some papers of Parshin, in which he describes some examples, like $PGL_2$ and $PGL_3$ over ...

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### Local Inverse Galois Problem

It's a basic fact that a finite Galois extension $L/K$ of a local nonarchimedian field $K$ has solvable (in fact supersolvable) Galois group $G$. One sees this by using the ramification filtration ...

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### Maximal separable extension of $\mathbb F_q((t))$

Let $K=\mathbb F_q((t))$. I want to prove that $K^{sep}$ is composite of $K^{sep}(p)$ and $K^{sep}(not \ p)$, where $K^{sep}(p)$ is maximal Galois extension of $K$ of exponent $p$, $K^{sep}(not \ p)$ ...

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### Global Artin reciprocity law from Local class field theory

Let $K=\mathbb F_q((t)), p -$ prime ideal in $K$, $\psi_p$ be the local Artin map$K_p^* \to Gal(K_p^{ab}/K_p)=G_p \subset Gal(K^{ab}/K)$. Then I define global Artin map $\psi_K$as product of $\psi_p$, ...

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### Reference for Local class field theory via witt vectors

I would like to find some books or lecture notes on geometric local class field theory via Witt vectors. I can't find any good paper on this subject.All approaches in the books to local class field ...

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### Valuations in Higher-dimensional local fields

I have the following question which I believ should be true but I would like to have a different opinion about it:
Let $M/L$ is a finite Galois extension of $n$-dimensional local fields and ...

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### Showing that $2c_1(f_*\mathscr O_X)=-f_*R_f$ on curves, maybe by local fields

I originally asked this question on Mathematic StackExchange, but it did not seem to be attracting any attention, so now I am trying mathoverflow. I hope it is not too simple or unappropriate a ...

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### Clarification about Tits' article in the Corvallis

I am studying Tits' article in the Corvallis wherein he defines the apartment in the general case (not necessarily split). I wish to know what he means about the filtration of the groups $U_a(K)$ ...

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### Ramified complete discrete valuation rings as extensions

Suppose $O$ is a complete discrete valuation ring with uniformizer $\pi$ and residue field $k=O/\pi O$ of charactersitc $p>0$. If $\nu$ is the $\pi$-adic valuation on $K=Frac(O)$, suppose also ...

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### Existence of maximal totally ramified $p$-extension of a local field

This relates to this question:
Existence of maximal totally ramified extensions of an arbitrary CDVF
Let $K$ be a local field with finite residue field of characteristic $p>0$. Does there exist a ...

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### Algebraic characters and quasi-characters of reductive algebraic group over non-archimedean local field

Let $G$ be a reductive algebraic group over $F$, where $F$ is a non-archimedean local field.
Then $G(F)$ is a p-adic group.
Let $\Psi(G)$ be the lattice of algebraic characters.
Let $\Lambda_G$ be the ...

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### Epsilon factors for tamely ramified extensions of local fields

Let $F$ be an unramified extension of $\mathbf{Q}_p$ of degree $n$, and let $K = F(\alpha)$ where $\alpha$ satisfies $\alpha^{p^n - 1} = -p$.
I'm interested in the local $\varepsilon$-factors ...

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### a question about a result in Bushnell-Henniart book 'the local Langlands conjecture for GL(2)'

This might be a easy question, but I couldn't get the point.
Let $F$ be a p-adic field, $\bar{F}$ a separable algebraic closure of $F$. Set $\Omega_F=Gal(\bar{F}/F)$. Use $F_{\infty}\subset \bar{F}$ ...

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### maximal abelian extension of quadratic extension of $\mathbb Q_p$

I read this article "Local class field theory via Lubin-Tate theory" http://arxiv.org/pdf/math/0606108v2.pdf. And I want to find the maximal abelian extensions for quadratic extensions of $\mathbb ...

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### Pro-$l$ Sylow action in a primitive representation of inertia over $\overline{\mathbb{F}}_l$

Let $K$ be a nonarchimedean local field of residue characteristic $p \neq l$ and let $I_K$ be the inertia subgroup of its absolute Galois group. Let $V$ an irreducible representation of $I_K$ over ...

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### Is $SL_1(D)$ toplogically finitely generated, for $D$ a division algebra over a local field?

I've been struggling with this one all day, and I was wondering if someone can give me a hand with the proof. I'm not even sure if the group in question is finitely generated, so I would appreciate if ...

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### Semisimple group not split by an unramified extension?

Let $F$ be a nonarchimedean local field. Does there exist a semisimple algebraic group over $F$ which is not split over a maximal unramified extension of $F$ ?

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### Reference request: expository text on the structure of reductive groups over non-archimedean local fields

I am interested in an expository text in English, which summarizes the main results and aspects of the structure theory of reductive groups over local fields, in a hopefully not very technical manner ...

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### Inertia group vs. differential equations

The tame quotient of the inertia group of $\mathbf Q_p$, say, is the profinite group generated by the Frobenius $\sigma$ and the monodromy $\tau$, subject to the relation $\tau^{p-1} [\tau, \sigma] = ...

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### A question on Haar measure on local field.

Let $F$ be a local field of characteristic 0, and $f:F\rightarrow \mathbb{C}$ be an integrable function. Is the following formulation valid?
$
\int_{F^\times}f(x^2) d^\times x=\int_{F^{\times 2}}f(x) ...

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### Maximal tamely ramified extension of $\mathbf Q_p$

Is there an explicit description of the maximal tamely ramified extension of $\mathbf Q_p$?
Thank you.

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### henselization and completion

This might not be a question appropriate for this forum, I apologize in this case...
Is it true that any element of the completion of a valued ring $R$ that is algebraic over the field of fractions of ...

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### What's the minimum number of generators for the wild inertia?

Suppose $K$ is a finite extension of $\mathbb{Q}_p$ and $K^{nr}$ the maximal unramified extension of $K$ in some fixed algebraic closure. Let $G_K$ be the absolute Galois group of $K$ and let $I_w$ be ...

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### Galois invariants in a ring of fractional power series over a finite field

Let $\mathbf{F}_q$ be a finite field, and let $A=\mathbf{F}_q [[ x^{1/q^\infty} ]]$ be the completion of $\mathbf{F}_q[x^{1/q^\infty}]$ with respect to the $x$-adic topology. Then the $q$th power ...

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### Cartan decomposition for upper triangular matrices

Due to the comments, I have the impression that I have to be more precise.
Consider $G= GL_n(F)$ for a non-Archimedean field $F$ with ring of integers $o$.
Let $K= GL_n(o)$ and let $I$ the Iwahori ...

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### Local densities of hermitian forms

I think this is an easy question, but I need some time to introduce it. I need to apply Yumiko Hironaka's computations on local densities of hermitian forms (see 1).
I would have liked to create the ...

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### How to understand the representation theory of $SL(n)$ from $GL(n)$?

Let $F$ be a local field. Consider the group extension (split)
$$ PSL(n,F) \rightarrow PGL(n,F) \rightarrow F^\times / (F^\times)^n.$$
What knowledge about $PGL(n)$ is necessary in order to understand ...

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### Automorphisms of local fields

It is an amusing coincidence (at least it appears to be a coincidence to me) that any completion of the field $\mathbb{Q}$ has trivial automorphism group as an abstract field, i.e. when ignoring the ...

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### squares in dyadic local fields

Hello,
By the local square theorem I know that $1+4\alpha$ is square if $|\alpha|<1$ ($\alpha$ is not a unit). Now, Can I always get a unit $\alpha$ such that $1+4\alpha$ is not a square ?? For ...