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Questions tagged [local-field]

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Additive-Group-Homomorphisms from a Local Field to the Non-zero complex numbers

I'm quite familiar with the $p$-adic numbers and $p$-adic analysis, so I already know that any continuous group homomorphism $\varpi_{p}:\left(\mathbb{Q}_{p},+\right)\rightarrow\left(\mathbb{C}\...
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1answer
67 views

Does $ \ (\pi \mathbb{Z}[\zeta_p])^2 \subset S \ \ $ hold ?, p-adic numbers

$\underline{\text{p-adic Numbers}}:$ Consider the cyclotomic extension $k=\mathbb{Q}_p(\zeta_p)$ of the p-adic field $\mathbb{Q}_p$ and let $ \mathbb{Z}_p[\zeta_p]$ be the ring of integer of $\mathbb{...
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2answers
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norms on a field and induced metric

Let $F$ be a field. A norm on $F$ is a map $|\,|:F\rightarrow\mathbb{R}$ with conditions: $|x|\ge 0$ and $|x|=0$ if and only if $x=0$. $|xy|=|x|.|y|$ $|x+y|\le |x|+|y|$ A norm $|\,|$ is said to be ...
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0answers
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Change of variables in $p$-adic integration

I'm trying to understand Serre's mass formula, but it's not very clear to me what does it mean to integrate over a subset of $L$, or $K^n$, and in particular why it holds an analogue of the usual ...
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Looking for the english translation of Serre's mass formula

I am looking for an English translation of the paper by Serre titled: Une "formule de masse'' pour les extensions totalement ramifiées de degré donné d'un corps local Could you please help me? ...
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1answer
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Continuity of product in absolute value

Let $F$ be a field, and $\| \cdot\|$ be an absolute value (or norm). I want to prove that with this norm, multiplication is continuous map on $F$ in the sense: for $x_0,y_0\in F$, and any $\...
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2answers
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Extension of degree 5 not obtained by adjoining a 5th root and whose normal closure contains a primitive 5th root of unity.

The question is mainly what the title says, but here is the setup in more details. Let $K$ be a field not containing a primitive 5th root of unity (for this question, the case $K = \mathbb{Q}$ seems ...
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1answer
38 views

Complete valuation, norm of finite extension. Proof of Propositon.

I would like to ask for tips how to manage with proof of this proposition. How to show that $v$ can be uniquely extended to $v'$ ? Should I assume that $v$ can be also extended to another valuation? ...
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Unramified extensions and base change

Let $K'/K$ be a finite extension of local discrete valued fields, and let $E/K$ be any extension of local discrete valued fields. Assume that $E\otimes_KK'$ is a field, and that $E\otimes_KK'/E$ is ...
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1answer
63 views

Meaning of Hasse-Arf theorem

I am reading about the Hasse-Arf theorem in Serre's 'Local Fields' and I have a hard time understanding what exactly it means for the upper numbering to have jumps only at integers. It seems like a ...
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1answer
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Extending absolute values on local fields - what is the 'correct' normalization and the relation to the global theory?

I'm having a tough time figuring out the 'correct' normalization for extending absolute values of local fields. I'm also trying to piece together how this interacts with the global theory, so below is ...
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Unramified extension of $K_π$ and $K_{π,n}$

I have some questions while reading Milne’s CFT. the fields below are assumed to be separable. Here $K$ is a local field,$π$ is a fixed prime element of $K$ and $K_π$ is the sub-field of $K^{ab}$ ...
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Indecomposable irreducible representation of $\mathrm{GL}(2, F)$ over local fields

Let $F$ be a local field and $\pi:F^{\times}=\mathrm{GL}(1, F)\to \mathrm{GL}(2, \mathbb{C})$ be indecomposable non-irreducible admissible (smooth) representation. Here indecomposable mean that it can'...
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2answers
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cyclic extension of prime power of a local field

Let $K$ be a non archimedian local field of characteristic $p>0$ with residue field $\mathbb{F}_p$ and $l\neq p$ be a prime. It is known by local classfieldtheory that any abelian Galois ...
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1answer
52 views

sum of sequences converging to $0$

Let $I$ a countable set of indices, and for any fixed $i\in I$ let $(a^{(n)}_i)_{n\in\mathbb N}$ be a sequence with values in a non-archimedean local field (e.g. $\mathbb Q_p$). Assume that the ...
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2answers
101 views

Good reference for local fields?

I learned and studied basic algebraic number theory (like number fields and extensions, prime decompositions, local fields, some of class field theory, ...) and I found that I'm not familiar with ...
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Lubin-Tate theory for non-maximal orders

Assume $O$ is an order in a $p$-adic field $K$, does there still exist a good theory of Lubin-Tate formal $O$-module? What field extension we will get by adding torsion points of such formal $O$-...
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What is wrong with my conclusion (compositum of local fields)?

Let $K = \mathbb{Q}_2(\zeta_3)$ where $\zeta_3$ is a primitive third root of unity, and $F = \mathbb{Q}_2(\zeta_3,\sqrt[3]{2})$. Furthermore, let $L = \mathbb{Q}(\zeta_3,\beta)$ where $\beta$ is ...
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1answer
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Computing degrees and ramification indices of some extensions of $\mathbb{Q}_2$

Let $K=\mathbb{Q}_2$ and $F = K(\zeta_3,\alpha)$ where $\zeta$ is a primitive third root of unity and $\alpha$ is a cubic root of $2$, i.e. $\alpha^3 = 2$.Let $K_1 = K(\zeta_3)$, $K_2 = K(\alpha)$ and ...
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1answer
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Proof of a Lemma for a local field extension with certain properties

The following result is from "Euler Factors determine local Weil Representations" by Tim and Vladimir Dokchitser: Lemma 1: Let $F/K$ be a cyclic extension of degree $n$ and ramification degree $e$. ...
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2answers
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Cyclic Field Extension of Local Field

Let $K$ be a local field (therefore complete, discrete non-archimedian valuation field) with perfect residual field $\kappa_K:= \mathcal{O}_K/\pi_K$. Assume that $L/K$ is a field extension of $K$ of ...
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Ramified Field Extensions

Let $k$ be a field of $char(k)=0$ und we consider an field exension $L/k$ with $[L:k]=n$. Set $M:= L((t^{1/n}))$ and $F:= k((t))$. I'm looking for a proof of following two statements: 1) If the ...
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Local fields. $p$-field $(k,v)$ Proof of lemma.

I study a lemma from 'Local Class Field Theory' and I have difficulties. To understand this properly we need another lemma Lemma:1 Let $\mathfrak{l=o/p}=\mathbb{F}_q$ for a $p$-field $(k,v)$. ...
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1answer
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Local fields. A $p$ - field $(k,v)$ with $\mathfrak{l=o/p}=\mathbb{F}_q$. Proof of lemma.

$(k,v)$ i a local field, $\mathfrak{p} = \{x | x \in k, v(x) > 0 \}$, $\mathfrak{o}=\{x | x\in k, v(x) \geq 0\}$. I'm working on Local Fields and I don't understand few things in proof of this ...
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2answers
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Construction of cyclic local field extensions of arbitrary degree and ramification index

Let $K$ be a local field. Let $n$ be an arbitrary natural number and $e$ be any divisor of $n$. Question Does there exist an extension $L/K$ with the following properties? $L/K$ is a cyclic ...
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0answers
50 views

Image of a character remains the same when restricting to a totally ramified extension

Problem I want to prove: Let $\chi: G_K \to \mathbb{C}^*$ be an unramified character and let $L/K$ be a cyclic totally ramified extension. Then $\chi(G_K)=\chi(G_L)$. All I managed to do was ...
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I got stuch finding Newton polygon of the following product with any easiest method

Find the Newton polygon of the following polynomials: $(i) \ f(X)=(1-X)(1-pX)(1-p^3X)$, $ (ii) \ g(X)=\prod_{i=1}^{p^2} (1-iX)$. Answer: $(i)$ To find the Newton polygon for the polynomial $f(X)$,...
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Newton polygon : Show that precisely $ l$ of the $ \lambda_i$ are equal to $ \lambda$

$\text{Newton Polygons for Polynomials}$ There is a lemma in the book $ \ \text{p-adic numbers, p-adic analysis and zet-functions} $ of the author $ \text{Neal Koblitz} \ $ which I mentioned below: $...
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1answer
63 views

Relation between local and global inertia/ramification degrees

Let $K/\mathbb{Q}$ be a number field and suppose a prime $p\in\mathbb{Z}$ factors in $\mathcal{O}_K$ as $\prod_{i=1}^r \mathfrak{p}_i^{e_i}$. From algebraic number theory, we have the identity $$ [K:\...
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1answer
89 views

Unramified subextension of the Galois closure of a totaly ramified $p$-adic field

Let $L/L'/K/\mathbb{Q}_p$ be a tower of finite field extensions such that $L'/K$ is totally ramified and $L/K$ is its Galois closure. We suppose that $L/L'$ is unramified. Let $M/K$ be a Galois ...
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1answer
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About the existence of tamely ramified extensions

I'm trying to understand the proof of the existence of tamely ramified extensions. For this, the theorem from my book says: Let $K$ be a complete field with respect to a discrete valuation, and let $\...
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1answer
159 views

Characterization of finite cyclic totally ramified extension of local fields with prime power degree

Definition Let $G_K$ be the absolute Galois group of a local field $K$. We will call a group homomorphism $\chi: G_K \to \mathbb{C}^*$ with finite image a character on $K$. Since every finite ...
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The local invariant map is a group homomorphism

Let $ (K, \nu) $ be a nonarchimedian local field. I have read that the Brauer group, $ \text{Br}(K) $ (which for me, is defined by the similarity classes of CSAs with group operation as tensor product)...
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1answer
110 views

Why can we always restrict an Galois representation so that it becomes unramified?

Let $K$ be a local field and $\rho: G_K \to \operatorname{GL}_n(\mathbb{C})$ be a Galois representation where $G_K$ denotes the absolute Galois group of $K$. We call a Galois representation $\rho$ ...
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1answer
40 views

Do the local polynomials of Weil representations coincide if they are Artin representation (factor through a finite quotient)?

Let $K$ be a local field, $G_K$ its absolute Galois group, $I_K$ the inertia subgroup of $G_K$, $\operatorname{Frob}_K \in G_K$ be a Frobenius element, i.e. any element of $G_K$ acting as $x \...
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Why are all Frobenius elements conjugated?

Let $K$ be a local field, $k$ its residue field and $G_K$, $G_k$ be the absolute Galois groups of $K$ and $k$, respectively. A Frobenius element is an element $\operatorname{Frob}_K \in G_K$ such ...
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How can one check if two totally ramified extensions of the same degree are equal?

Let $F/K$ and $L/K$ be finite and totally ramified extensions of local fields whose degree are equal. Question: How can we check if $F=L$ or not? My thoughts and attempts: If we look at $K=\mathbb{...
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Can one write a finite extension of local fields as a compositum of fields whose degrees are prime powers?

Let $F/K$ be a finite extension of local fields of degree $n$. Question: Does there exist intermediate fields $F/K_i/K$ such that the degree of $K_i/K$ is a prime power and $F$ is the compositum of $...
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2answers
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Totally Ramification and Quadratic extension of p-adic field $ \mathbb{Q}_p$

$ \text{Totally Ramification}:$ Consider the quadratic extension $K=\mathbb{Q}_p(\sqrt 3)$ of the p-adic field $\mathbb{Q}_p$. Here, $$ K=\mathbb{Q}_p(\sqrt 3)=\{a+b \sqrt 3: \ a,b \in \mathbb{Q}...
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Do elements having any $p$ power roots form a open subset in a perfectoid field

Let $K$ be a perfectoid field, if $K$ has positive characterestic then every $a \in K$ has any $p$ power roots i.e $a=x^{p^n}$ have solutions for any positive integer $n$. However, in characteristic ...
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2answers
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Under What assumptions on $p$, $\mathcal{O}_K^* \simeq \mathbb{Z}_p^{*} \oplus \mathbb{Z}_p^{*}$

Let $p$ be a fixed prime number and $\mathbb{Q}_p$ be the field of $p$-adic numbers and $K$ be an extension of degree $2$ of $\mathbb{Q}_p$. Let $\mathcal{O}_K$ be the ring of integers of $K$ and $\...
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1answer
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Why is the local polynomial of an Artin representation independent of the choice of the Frobenius element?

I will take the notation from this paper on which I am currently working. Let $K/\mathbb{Q}_p$ be finite (i.e. K is a local field) and $F/K$ be a Galois extension. We shall denote by $\mathbb{F}_K$ ...
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Is the Intertia subgroup the whole Galois group if our finite extension of local fields is totally ramified?

Let $F/K$ be a finite extension of local fields which is totally ramified. The inertia subgroup is defined as $$ I_{F/K} = \{ \sigma \in \operatorname{Gal}(F/K) : \sigma(x) \equiv x \mod \pi_F \: \...
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Definition of the Weil group: Question about exact sequence with Inertia Group and absolute Galois group over a local field

Let $K$ be a local field, $k$ be its residue field, $G_K, G_k$ be the absolute Galois groups of $K, k$ and $I_K$ be the inertia group of $K$. In several books and papers, I found the following exact ...
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0answers
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Vanishing of second Galois cohomology group

This most likely follows from a standard result but a lack of knowledge prevents me from seeing this. Let $K$ be a non Archimedean local field. Let $\Gamma$ be $\mathrm{Gal}(\bar{k}/k)$. Let $T$ be a ...
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1answer
53 views

When is an element central in the image of a Weil representation?

While reading Lemma 3 of this paper, I encountered the following statement: Take a sufficiently large finite Galois extension $F/K$ such that $\rho/F$ is unramified. Then $\rho(\operatorname{Frob}...
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1answer
93 views

Non-Galois finite extensions of $F^{\operatorname{ur}}$

Let $F$ be a $p$-adic field with algebraic closure $\overline{F}$, and let $F^{\operatorname{ur}}$ be the maximal unramified extension of $F$. Let $E$ be a finite extension of $F$. I'm a bit rusty ...
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37 views

Is this subset dense in a complete field?

Let $(K,v)$ be a field equipped with a non-archimedean valuation $v,$ and let $(L,w)$ be a finite extension of $(K,v)$ with $w$ of course non-archimedean. If we denote by $K_v,L_w$ the completions of $...
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1answer
23 views

Open subgroup of $ \mathcal{O}_L $

Let $ L/ K $ be a finite Galois extension with $ K $ a local field and $ G $ its Galois group. By the normal basis theorem, there is a normal basis $ \{ \sigma_i(x) : \sigma_i \in G \} $, moreover, by ...
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1answer
53 views

Question regarding the statement of the Local Existence Theorem in Local Class Field Theory

Let $K$ be a non-archimedean local field. Then the local existence theorem states that the norm groups in $K^*$ are exactly the open subgroups of finite index. Here what is meant by an 'open' ...