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.

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Soluions over ring of integers from an non archimedean field

This is Proposition 1.9. I am confused with the forward direction: If $K$ is alg. closed narc field with $x$ a nonzero element with posiive norm. Then: $K$ is not discrete. Every polynomial over $...
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1answer
20 views

Cyclic totally ramified Galois extension of of non-archimedean fields

Let $K \subset L := K(a)$ be a simple totally ramified extension of non-archimedean local fields of degree $n$ generated by a $n$-th root of $K$; ie $a$ is a root of irreducible polynomial $X^n- b \in ...
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1answer
36 views

Pontryagin dual of the multiplicative group of a local field.

Let $K$ be a local non-Archimedian field. Let $K^{\times}$ be the group of invertible elements of $K$. Is there an explicit description of the Pontryagin dual of $K^{\times}$?
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Bernstein center

I am trying to read the lecture notes of Joseph Bernstein on the representations of p-adic groups and struggling to understand a certain claim regarding the Bernstein center. suppose G is a reductive ...
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3answers
54 views

Can a generator of the ring of integers of local fields can be chosen so that it is also a uniformizer at the same time?

Let $L/K$ be an extension of local fields. We can find $\alpha$ such that $\mathcal{O}_L=\mathcal{O}_K[\alpha]$. What do we know about this generating element? I think that this $\alpha$ can be ...
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1answer
19 views

basic example of unramified extension by take a root of a unit (not necessarily 1)

Let $K$ be a local field with residue characteristic $p$, and let $u \in O_K$ be a unit. Let $e$ be relatively prime to $p$. Let $L=K(u^{1/e})$ why is $L/K$ an unramified extension? Do we need $(e ,p)=...
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1answer
28 views

Functoriality of the maximal unramified extension

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $L$ a finite Galois extension of $K$. We have $$\text{Gal}(K^{nr}/K)\cong\hat{\mathbb{Z}}\cong \text{Gal}(L^{nr}/L) $$ where $K^{nr},L^{nr}$ denotes ...
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21 views

Local function inversion

I'm looking for a reference dealing sith local function calculus. Up to now I believe the following : Suppose a local function $y=f_\bar{x}(x)$ were given by $y=\bar{x}x$. Is the inversion given by : $...
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0answers
53 views

Reference Request — Galois theory of local fields

I've been trying to read the following note: http://bicmr.pku.edu.cn/~lxiao/2020fall/Lecture1.pdf On the page 5 - 6, there is a brief review on the Galois theory of local fields. I'm trying to ...
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1answer
46 views

Functoriality of the local Artin reciprocity

Let $k$ be a finite extension of $\mathbb{Q}_p$. Then the local Artin reciprocity tells us that $$\text{Gal}(\bar{k}/k)^{\text{ab}}\cong \hat{k^\times}.$$ If we have a finite extension $\bar{k}/l/k$, ...
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27 views

The equation $uX^p+X+d^p=0$ in $k((x))$

Let $k$ be a finite field in characteristic $p$ and let $K$ be the field $k((x))$. Let $u$ be a unit of $k[[x]]$ and let $d\in K$ but not in $k[[x]]$. Does the polynomial $uX^p+X+d^p$ have a roots in $...
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46 views

Is the ugly lemma enough for local class field theory?

From what I understand, it follows from the ugly lemma (page 135 Cassels and Frohlich) that $H^2(Gal(K^{un}/K), {K^{un}}^{\times}) \cong H^2(Gal(K^{sep}/K), {K^{sep}}^\times) $. One can then define ...
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2answers
70 views

Galois action on $K\otimes_{\mathbb{Q}} \mathbb{Q}_\ell$

Let $K/\mathbb{Q}$ be a finite extension and let $\ell$ be a prime number. For each prime $\lambda$ of $K$ lying over $\ell,$ choose an embedding $\sigma_\lambda:K\hookrightarrow K_\lambda$ of $K$ ...
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1answer
73 views

Canonical homomorphism between $K/\mathcal O_K$ and $S^1$

Is well know that there a isomorphism 1. $$\mathbb Q_p/\mathbb Z_p\approx \mathbb Z[1/p]/\mathbb Z\hookrightarrow \mathbb R/\mathbb Z\approx \mathbb S^1.$$ Now, let $K/\mathbb Q_p$ be a finite ...
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0answers
51 views

Free finite index submodule of $\mathcal{O}_L$

Let $L/K$ be a finite Galois extension of $p$-adic fields, with Galois group $G$. I've read somewhere that it is well-known that $\mathcal{O}_L$ contains a free finite index $\mathcal{O}_K[G]$-module, ...
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1answer
28 views

The correspondence of topologies of different definitions of local fields?

I'm recently looking about the classification theorem of local fields, starting from the definitions. One of the common definition is locally compact Hausdorff non-discrete topological field. The ...
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1answer
41 views

Finite quotients of ring of integers of local field

Let $K$ be a non-Archimedean local field, so either a finite extension of $\mathbb{Q}_p$ or a finite extension of $\mathbb{F}_q((t))$. Let $\mathcal{O}$ denote its ring of integers and $\pi$ a ...
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1answer
56 views

Abelian Varieties and Neron Models over local fields: Quotient of points of k mod valuation ring is finite

I am working on a paper and I am not sure why the following fact is true. Suppose A/k is an abelian variety over a nn-archimedean local field k and $N/\mathcal{O}$ should be the Neron model. Then let $...
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1answer
41 views

Formal power series on discrete valuation rings

Let's start with a simple result: For $F(x)\in \mathbb{Z}_p[x]$, $$F(x)^p\equiv F(x^p) \mod p $$ I generalize this result to a problem: Let $K$ be a local field, $A$ be its ring of integers with ...
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1answer
54 views

Decomposition of abelian closure of local field as $K^{ab }=K_{\pi}K^{unr} $

$K$ is a local field, we write $K^{\times}=U_K . \pi^{\mathbb Z} $ where $U_K$ is group of units of $\mathcal O_K $ and $\pi $ is the uniformizer of $K$. I wish to prove that $K^{ab }=K_{\pi}K^{unr} $ ...
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1answer
39 views

Quaternion algebras over a non-Archimedean local field $K$, up to isomorphism

I want to know the number of non-isomorphic quaternion algebras over a non-Archimedean local field $K$. What is the number of non-isomorphic central simple algebras of dimension $n^2$ over a non-...
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36 views

Showing that the additive group of a local field is locally profinite

Let $F$ be a non-archimedean local field, $\mathcal{O}$ its integers, $\mathfrak{p}$ its maximal ideal and $k := \mathcal{O}/\mathfrak{p}$ its residue field with $|k| = q$. Let $\pi$ be a uniforimser ...
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1answer
21 views

Countable basis of compact open subgroups in locally profinite group.

I am currently reading through a proof of the existence of a right haar measure for locally profinite groups. We have a locally profinite group $G$ with the assumption that for all compact open ...
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0answers
33 views

Artin conductor of cyclotomic extensions.

Let ${\Bbb Q}_p(\zeta_{p^n})/{\Bbb Q}_p$ be a cyclotomic abelian extension of degree $p^n - p^{n-1}$ and define $G \colon= {\mathrm{Gal}}({\Bbb Q}_p(\zeta_{p^n})/{\Bbb Q}_p)$, for which we set $G = \...
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1answer
60 views

Example of non Galois extension of a local field and its Galois closure

I am trying to think of an example of non Galois extension of a local field and its Galois closure. I started by looking at examples of extensions of $\mathbb Q_p $. For example I think $x^3-3$ is ...
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2answers
23 views

Examples of local ring with principal group units $U_{1}=U_{1}^{2}$

Let $A$ be a local ring with maximal ideal $M$. if we consider $U_{1}=1+m$. I am trying to find some examples of local rings where the condition $U_{1}=U_{1}^{2}$ holds. I was thinking like local ...
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0answers
15 views

How to decompose a multivariable power series into product of single variable power series?

Is there any known result of decomposing multivariable power series over local field or any field into product of single variable power series ? For example, consider the following power series in $n$ ...
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1answer
65 views

Is the extension $\text{Totally Ramified}$?

$(1)$ Let $K/F$ be a finite extension of the local field $F$ of characteristic $0$ obtained by adjoining by the roots of a irreducible monic polynomial to $F$. Is the extension $\text{Totally Ramified}...
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2answers
83 views

Examples for completions of number fields

i'm currently learning for an exam on class field theory. The first thing i thought about are examples for completions of number fields (here $K$), for example of the field extensions $\mathbb{Q}[\...
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1answer
76 views

Step in proof of Hasse-Arf theorem on ramification groups

This question concerns Yoshida's proof of the Hasse-Arf theorem in https://arxiv.org/abs/math/0606108 (page 16). For a totally ramified extension $K^\prime/K$ of local fields define the ramification ...
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1answer
41 views

Open subgroup of ring of ring of integers

I am trying to understand following lemma from Milne's Class Field Theory: https://www.jmilne.org/math/CourseNotes/CFT.pdf#X.3.2.3 (the link will take you directly to the said lemma) Let $L$ be a ...
3
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1answer
64 views

Equality of fields being deduced from isomorphism of Galois groups

I'm reading and attempting to understand a proof of the local Kronecker-Weber theorem in https://arxiv.org/abs/math/0606108 (page 17). Let $K$ be a local field and $\sigma \in W(K^{LT}/K)$ with $\...
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0answers
65 views

What does it mean to say that $G$ has a 'canonical topological generator' $\text{Frob}_{L/K}$?

$G$ is $\text{Gal}(L/K)$ where $L$ is unramified extension (could be infinite) of a local field $K$. What does it mean to say that $G$ has a 'canonical topological generator' $\text{Frob}_{L/K}$? Does ...
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1answer
82 views

Structure of unramified extension of $\mathbb{Q}_2$

Let $K$ be the unique unramfied extension of degree $2$ over $\mathbb{Q}_2$. In this case a well-known theorem, says that $\mathcal{O}_K^\times \cong \mathbb{Z}/(q-1)\mathbb{Z} \times \mathbb{Z}/2^a \...
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0answers
16 views

Periodic discrete linear groups over local fields

Let $\mathbb{K}$ be a non-Archimedean local field. I am interested in (infinite) periodic discrete subgroups of the locally compact group $GL_n(\mathbb{K})$. By a periodic group I mean a group in ...
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2answers
68 views

The closure of a number field under the $p$-adic topology

We fix an inclusion $\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}_p}$. Given $K$ a finite extension of $\mathbb{Q}_p$, can we always find $K_0$ a finite extension of $\mathbb{Q}$ such that ...
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1answer
46 views

Orders in extension of $Q_p$

Suppose we have field extensions $L/K/\mathbb{Q}_p$. If $R$ is a ring such that $$\mathcal{O}_K \subset R \subset \mathcal{O}_L$$ (where if $M$ is an extension of $\mathbb{Q}_p$ then $\mathcal{O}_M$ ...
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1answer
42 views

Residue field of an infinite extension of $\mathbb{Q}_p$

Let $\zeta_1 \in \overline{\mathbb{Q}_p}$ such that $\zeta_1^p=p$, now let $\zeta_2 \in \overline{\mathbb{Q}_p}$ such that $\zeta_2^p=\zeta_1$ and so on with $\zeta_i^p = \zeta_{i-1}$. I have to show ...
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1answer
28 views

Why this extension is purely inseparable.

Let $E$ be a local field of characteristic $p>0$. Let $\varphi$ be the absolute Frobenius map on $E$. Then how to prove that the field extension $E/\varphi(E)$ is purely inseparable.
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2answers
43 views

Any compact subset of $\bar{\mathbb{Q}}_{p}$ is in a finite extension of $\mathbb{Q}_p$?

It is well-known that any compact subgroup of $\bar{\mathbb{Q}}_{p}$ (algebraic closure of $\mathbb{Q}_p$) is in a finite extension of $\mathbb{Q}_p$. (Usually, its proof requires Baire category ...
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1answer
150 views

Isomorphic extensions of $\mathbb Q_p$

Suppose you are given two cubic Eisenstein polynomials in $\mathbb Q_p[X]$, say $f(X)=X^3-p$ and $g(X)=X^3-pX^2-p^2X-p$. If $\alpha$ is a root of $f(X)$ and $\beta$ is a root of $g(X)$, then the ...
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0answers
56 views

Checking if the intersection of two cyclic $p$-adic extensions with certain properties is trivial

Let $L$ and $L'$ be finite extensions of $K = \mathbb{Q}_p$. Also, let $n = [L:K]$ and $e = e(L/K)$. Furthermore, we assume the following properties: $L$ and $L'$ are both cyclic over $K$, $L'/K$ is ...
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1answer
51 views

Structure of the units of the valuation ring of a finite extension of $\mathbb{Q}_p$

Let $K$ be a finite extension of the field of $p$-adic numbers $\mathbb{Q}_p$, call $\mathcal{O}_K$ the valuation ring of $K$ (i.e. the set of elements of $K$ with valuation greater or equal than zero)...
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2answers
54 views

$K^{*n}$ has finite index in $K^*$ for an infinite extension field $K$ of $\mathbb{Q}_p$?

Let $K\subset\mathbb{C}_p$ be an extension field of $\mathbb{Q}_p$, then when $K/\mathbb{Q}_p$ is a finite extension, then $K^{*n}$ has finite index in $K^*$(see this question, we can decompose $K^*$ ...
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0answers
48 views

Lift Frobenius generator of Galois group $Gal(l/k)$ of an unramified extension $L/K$

We consider a finite unramified Galois field extension $L/K$ of non-Archimedean local fields with finite residue fields $l / k$. firstly, some notations: denote by $q \in O_K$ a uniformizer of $O_K$ ...
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1answer
90 views

A theorem of Lutz about the structure of the points of an elliptic curve over a finite extension of $\mathbb{Q}_p$

Reading the article of Greenberg "Iwasawa Theory for Elliptic Curves", he cites (p.13) a theorem of Lutz that says: Theorem: Let $E/K$ be an elliptic curve defined over a finite extension $K$ of $\...
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1answer
47 views

Algebraic closure of Puiseaux field $K((T))$ equals $\bigcup_{n \ge 1} K((T^{1/n}))$

I want to show that the algebraic closure $L:= \overline{K((T))}$ of Puiseaux field $K((T))$ for $K$ alg. closed of char $K=0$ equals the union $\bigcup_{n \ge 1} K((T^{1/n}))$. The closure clearly ...
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1answer
103 views

Reference for properties of absolute Galois group of local field

Let $K$ be a local field. Let $K^{nr}$ and $K^t$ be its maximal unramified and tamely ramified extensions, respectively. One can show that $\operatorname{Gal}(K^{nr}/K) \cong \widehat{\mathbb Z}$ and ...
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0answers
23 views

ord map from a local field to $\mathbb{Z}$

I'm reading J. Milne's notes on class field theory and near in the section of the chapter on local CFT he speaks of an exact sequence $$1 \to U_L \to L^{\times} \xrightarrow{\textrm{ord}_L} \mathbb{Z} ...
2
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1answer
179 views

Galois group of tamely ramified extension

Let $p$ be a prime and let $K$ be a finite extension of $\mathbb{Q}_p$. Suppose $L/K$ is a tamely ramified Galois extension. I want to show that if $\sigma$ is a lift of the Frobenius element of the ...

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