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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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Examples of Lubin-Tate formal groups

I'm currently learning about Lubin-Tate theory: Given a p-adic field $K$ and a uniformizer $\pi \in \mathcal{O}_K$, they consider formal $\mathcal{O}_K$-modules $F$ such that the endomorphism $[\pi]$ ...
Fungaria's user avatar
1 vote
1 answer
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Inertia field example of $ \mathbb{Q}_5(\sqrt[4]{50})$

Let $L = \mathbb{Q}_5(\sqrt[4]{50})$ and denote by $E$ the inertia field of the extension $L / \mathbb{Q}_5$. Write down a prime element $\pi_E $ of $ \mathcal{O}_E $ with $L = E(\sqrt{\pi_E})$. Can ...
Christian Schwacke's user avatar
1 vote
0 answers
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Relating discriminants of hyperelliptic curves to discriminants of their defining polynomials

Let $C$ be a hyperelliptic curve defined by an equation of the form $$ C: y^2=f(x) $$ where $f$ is a polynomial of prime degree $p\geq3$, over a complete field $K$ of residue characteristic $p$. ...
did's user avatar
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1 answer
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could Eisenstein polynomial admit multiple root?

That is the question ! In a local field could an Eisenstein polynomial admit a mutiple root ? A point in the construction of Lubin-tate extension seems to need only simple roots. Thank you
noradan's user avatar
  • 309
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1 answer
48 views

Possibly inseparable extensions of $\mathbb{F}_p((t))$

I have a question on local fields. Some sources define a characteristic $p$ local field as of the form $k((t))$ where $k/\mathbb{F}_p$ is a finite extension. Some sources define it as a finite ...
Kai Wang's user avatar
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3 votes
0 answers
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Norm surjective for unramifeid extension of local fields $L/K$

I have a question about the argument used here in proof of Proposition 4.2.5.: For any finite unramified extension $L/K$ of local fields, the map $\text{Norm}_{L/K}: O_L^* \to O_K^*$ is surjective. ...
user267839's user avatar
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1 answer
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Understand the first unit group $U^{(1)}$ as a $\mathbb{Z}_p$-module

Let $K$ be a finite extension of $\mathbb{Q_p}$. It is well-known that $K^\times$ is isomorphism to $\mathbb{Z}\times\mathbb{Z}/(q-1)\mathbb{Z}\times\mathbb{Z}/p^\alpha\times\mathbb{Z}_p^d$, where $q$ ...
RoastedFish's user avatar
1 vote
0 answers
43 views

Silverman AEC Exercise 7.4

The following problem in chapter 7 of Silverman's book has been bothering me: Here's my progress: I first tried to solve it in the $\text{char}(k) \ne 2,3$ case, wherein we can assume the equation is ...
Aditya Khurmi's user avatar
3 votes
0 answers
32 views

Is there any relationship between the study of class number of a field and the study of class field theory through Lubin-Tate formal group?

I am curious to know if one can relate to the study of Lubin-Tate formal group and related local class field theory with the study of class number of a field (global field in general). As far as I ...
MAS's user avatar
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Norm $\text{N}_{L/ \Bbb Q_p}(\pi_L)$ of Uniformizer of finite extension of $p$-adics

Let $ K/ \Bbb Q_p$ a finite extension of $p$-adic field $\Bbb Q_p$ of degree $[K:\Bbb Q_p]=n$ and let $ \pi_L$ be a uniformizer of $L$. Question: Can we say something interesting / "distinguished&...
user267839's user avatar
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1 vote
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Upper estimation for Multiplicative Conductor of Extension of $\Bbb Q_p$

Let $L/\Bbb Q_p$ be a finite extension of $p$-adics of degree $n$. Let $f$ be the minimal number such $1 +(p)^f:=U^f \subset \text{Norm}_{L/ \Bbb Q_p}(L^*)$. The ideal $(p)^f$ is also called ...
user267839's user avatar
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Computation of Norm

I am attempting an exercise from a Galois Theory problem sheet I found online. It asks: Let $p$ be an odd prime. Let $L_1 = \mathbb{Q}_p(\zeta_p) / \mathbb{Q}_p$ and let $L_2=\mathbb{Q}_p(\sqrt[p-1]{-...
Todd Burnett's user avatar
2 votes
1 answer
41 views

A limit problem in $p$-adic fields

I met a very concrete limit computation while dealing with some formulas in $p$-adic analysis. Let $F$ be a finite extension over $\mathbb{Q}_p$ with $p>2$ a prime. Let $E/F$ be a ramified ...
youknowwho's user avatar
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1 vote
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Example of extension of a non-Archimedean local field where the inertia subgroup is not open

I want to to find an example of an algebraic extension L of an non-Archimedean local field K where the inertia subgroup is not open when viewed in Gal(L/K). First thing to note is that we need the ...
Rick's user avatar
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3 votes
1 answer
116 views

Field extension $K/\Bbb{Q}$ of given $e,f,g$ .

Let $e,f,g$ be an fixed arbitrary positive integers. I want to find an example of prime number $p$ and a degree $efg$ algebraic extension $K/\Bbb{Q}$ whose ramification index is $e$,inertia degree is ...
Poitou-Tate's user avatar
  • 6,353
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0 answers
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Passing from global number field to local number field

I want to clarify my understanding of the localization of the global field. Let $K$ be a number field, that is a finite extension of $\mathbb Q$ or $\mathbb{F}_p(x)$. These are not complete fields. ...
MAS's user avatar
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2 votes
1 answer
40 views

Scheme theoretic reduction map for elliptic curves

Consider $K$ a local field with $\text{char }k=0$ and $A$ an abelian surface over $O_K$. Let $A[2]$ be the 2-torsion finite flat group scheme over $O_K$ and $A[2]^\circ$ be the connected component of ...
Ja_1941's user avatar
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1 vote
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$\mathrm{GL}_n(\mathbb Q_p)$ is unimodular

I'd like to prove the well-known result that $G = \mathrm{GL}_n(\mathbb Q_p)$ is unimodular, using elementary results, i.e. without reductive groups. Some definitions: as a locally compact group, $G$ ...
Gargantuar's user avatar
2 votes
0 answers
21 views

Norm and residue field algebraically closed

$K$ is a dv field with residue field algebraically closed and $L$ a finite extension I read as "well known" that the norme is onto on units. I known this result for unramified extension but ...
noradan's user avatar
  • 309
3 votes
1 answer
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A local sort-of Dedekind criterion

In class I have seen that for $L/K$ a finite extension of number fields with $L=K(a)$ where $a$ has min poly $f$. Then for a prime ideal $\mathfrak{p}$ we have that $$L \otimes K_\mathfrak{p} \cong \...
Rick's user avatar
  • 391
0 votes
0 answers
45 views

Some questions about almost étale extension of local field

I'm in trouble understandig the example and proof of Example 4.2.2 in Denis Benois's lecture note about $p$-Adic Hodge Theory, he gives the following definition: A finite extension $E/F$ of non-...
Kevin's user avatar
  • 395
3 votes
0 answers
64 views

$\mathfrak{p}$-adic valuation of a norm

Given extension of Number fields L/K and prime ideal $\mathfrak{P} \in O_L$ lying over $\mathfrak{p} \in O_K$. Let $\hat{L}$ and $\hat{K}$ be the completions of L and K respectively. And let e and f ...
Rick's user avatar
  • 391
0 votes
2 answers
44 views

Set of representatives vs Teichmüller representatives

Say $K$ is a local field (complete wrt a discrete valuation with finite residue field, or if you want perfect residue field). Denote its residue field by $k$ where $k= \mathcal{O}_K / \mathcal{M}_K.$ ...
berightback's user avatar
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0 answers
19 views

problem with local Artin map in Hazewinkel article

$\def\cU{{\cal U}} \let\lra\longrightarrow$ I'm studying the Hazewinkel article "Local class field is easy" (a bit presomptuous as far as i'm concerned !). Readable here https://www....
noradan's user avatar
  • 309
1 vote
1 answer
102 views

Is the $\mathbb F_{p^n}((x,y))$ a local field?

Let $\mathbb{F}_{p^n}$ be a finite field of characteristic $p$, then we can define discrete valuation $v_p$ in $\mathbb{F}_{p^n}((x))$ by $$ v_p(f):=\min \{n~|~a_n \neq 0\}, $$ where $f(x)=\sum_{n \...
MAS's user avatar
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2 votes
1 answer
56 views

Matrices preserving the sup-norm on $\mathbb{Q}_p^n$

Consider the $p$-adic field $\mathbb{Q}_p$ and $n\geq 1$. The absolute value is $|x|=p^{-v_p(x)}$ for $x\in\mathbb{Q}_p$, where $v_p$ is the $p$-adic valuation. Endow $V=\mathbb{Q}_p^n$ with the sup-...
Object's user avatar
  • 339
3 votes
1 answer
49 views

One dimensional $\ell$-adic representations of $\mathbb{Z}_p$

I am reading the paper "Lectures on the Langlands program and conformal field theory" by E.Frenkel. In page 28, he computes the one-dimensional continuous representations of $\mathbb{Z}_p$ ...
Runner's user avatar
  • 145
1 vote
1 answer
30 views

morphism on complete filtered group

I am studying complete local fields. For the third time i read proofs which seem similar. There is an application defined on a complete group with a filtration $F_n$ (i'm not sure if it's the right ...
noradan's user avatar
  • 309
0 votes
0 answers
20 views

Serre's definition of $U_{\mathfrak{m}}$ for $\ell$-adic representations

In Serre's "Abelian $\ell$-adic representations and Elliptic curves", in order to define the set $S_{\mathfrak{m}}$ he defines of $U_{v,\mathfrak{m}}$ as follows: $$U_{v,\mathfrak{m}}=\left\{...
Yang Awotwi's user avatar
0 votes
1 answer
74 views

Tamely extension of the maximal unramified extension of a local field

This construction comes from the chapter VII of "the local Langlands conjecture for $GL(2)$": Let $F$ be a local field. We know that for any $n \in \mathbb{N}$ there exists a unique ...
Mario's user avatar
  • 739
1 vote
1 answer
66 views

Elliptic curves - finding which primes $p$ guarantee there are $x,y\in\mathbb{Z}_p$ on the curve

For which primes $p$ do there exist $x,y\in\mathbb{Z}_p$ such that $3y^2=4x^3-10?$ This is a question in an elliptic curves course so I assume we want to transform this into an elliptic curve! My ...
user avatar
1 vote
0 answers
65 views

Why is the valuation of a separable closure equal to $\mathbb{Q}$?

I am reading a paper by Serre, in which at some point he say We extend the valuation $v$ from $K$ to $K^{sep}$; in this way we obtain a valuation on $K^{sep}$ with value group $v(K^{sep \times})=\...
Batrachotoxin's user avatar
1 vote
1 answer
30 views

galois group of completions

Let "ur" be for "unramified maximal". I read as "well known" that if $L/K$ is an abelian totaly ramified extension of the local field $K$ then $Gal(\widehat{L_{ur}}/\...
noradan's user avatar
  • 309
0 votes
0 answers
12 views

abelian extension of local field is unramified

Why is this result true : $K$ being a local field with algebraicaly closed residue field then an abelian extension of $K$ (let's say $L$) is totaly ramified! I didn't find any reference in all the ...
noradan's user avatar
  • 309
1 vote
1 answer
60 views

If $\mathbb Q_p(\sqrt[3]{2})$ is galois over $\mathbb Q_p$ then does $\mathbb Q_p$ necessarily contain a cube root of unity?

Denote by $\omega$ any non-trivial cube root of unity. If $\mathbb Q_p(\sqrt[3]{2})$ was galois then it contains all cube roots of $2$ i.e. $\sqrt[3]{2}$, $\omega\sqrt[3]{2}$ and $\omega^2\sqrt[3]{2}$ ...
Tony Pizza's user avatar
0 votes
0 answers
28 views

associate a character to an abelian extension in local class field theory

Let $K_v$ a local and $F$ a finite abelian extension of it. Two questions: Could somebody explain how local class field theory associates naturally a character of $K_v^{\times}$ to $F$. And, if $r \in ...
JackYo's user avatar
  • 179
0 votes
1 answer
83 views

embed roots of unity from valued field into its residue field

Let $K_v$ non-archim valued complete local field with finite residue field $\kappa= \mathcal{O}_K / \mathfrak{m}_v$ of characteristic $p$. Assume $K_v$ contains $d$th roots of unity $U_d$. Is there ...
JackYo's user avatar
  • 179
0 votes
0 answers
30 views

local fields are locally profinite

Let $F$ be a local field with ring of integer $\mathfrak{o}$ and maximal ideal $\mathfrak{p}$. It is clear that $$\mathfrak{o}\supseteq \mathfrak{p}\supseteq \mathfrak{p}^2\supseteq \dots $$ is a ...
Mario's user avatar
  • 739
0 votes
0 answers
103 views

Elliptic curve theory : Injection from $\widehat{E(K)}$ to $E(K_v)$

Let $E/K$ be an elliptic curve over a number field $K$. Let $\widehat{E(K)}$ be profinite completion of $E(K)$.// Let $v$ be a place of $K$. Let $K_v$ be completion of $K$ at $v$. Is there an ...
Poitou-Tate's user avatar
  • 6,353
1 vote
1 answer
16 views

proof of existence of totally ramified extension

I have a problem with the proof of Hazewinkel in his "Local Class Field" article. First I don't understand why is $F"$ well defined ? I think it needs $L\cap K_r=K$ but I don't see why it ...
noradan's user avatar
  • 309
0 votes
1 answer
40 views

density of $\langle \textit{Frobenius}\rangle$ in $Gal(L/K)$

$K$ is a local field. $L/K$ is Galois with $K_{ur}\subset L$, where $K_{ur}$ is the maximal unramified extension of $K$. Noting $\Phi\in Gal(L/K)$ a lift of the Frobenius $\phi\in Gal(K_{ur}/K)$ and $...
noradan's user avatar
  • 309
0 votes
0 answers
34 views

Construction of the Adelic Metaplectic Group

I'm essentially looking for a reference for the following statement on the wikipedia: It can be proved that if F is any local field other than $\mathbb{C}$, then the symplectic group $Sp_{2n}(F)$ ...
Riley Moriss's user avatar
0 votes
0 answers
33 views

The index of multiplicative group of local field

For a local field $K$, I know that $c:=(K^* : (K^*)^m) = mq^{v(m)}\mid\mu_m(K)\mid$, where $q$ is the number of residue field of $K$. I want to know the quantity $c$ when $m=2$. Using the above ...
WHERE 234's user avatar
  • 115
4 votes
0 answers
105 views

Show by Local Class Field Theory $\mathbf Q_p$ has unique Galois ext. iso. to $(Z/2Z)^2$ if $p > 2$, and unique Galois ext. iso. to $(Z/2Z)^3$ o/w.

Show that $\mathbf{Q}_p$ has a unique Galois extension isomorphic to $(Z/2Z)^2$ if $p > 2$, and that $\mathbf{Q}_2$ has a unique Galois extension isomorphic to $(Z/2Z)^3$ I have already completed ...
Rick's user avatar
  • 391
1 vote
2 answers
153 views

Can a local field be local in more than one way?

A local field (to my knowledge) is a field equipped with a non-trivial absolute value that makes it a locally compact topological space. From this, one derives that the local fields are $\Bbb R,\Bbb C,...
Hilbert Jr.'s user avatar
  • 1,434
0 votes
2 answers
77 views

Is maximal ideal of $\mathfrak{p}$ adic number field ideal extension of $\mathfrak{p}$?

Let $F_{\mathfrak{p}}$ be a completion of a number field $F$ at a non zero prime ideal $\mathfrak{p}$ of integer ring $\mathcal{O}$ and $\hat{\mathcal{O}}$ be its integer ring and $\hat{\mathfrak{p}}$ ...
user682141's user avatar
  • 1,016
1 vote
0 answers
60 views

Completion of a non-Galois extension

The discussion in these posts, Completions of number fields and Corresponding local extension does not depend on choice of prime ideal above?, gets at the fact that if $L/K$ is a Galois extension of ...
user10039910's user avatar
1 vote
0 answers
60 views

Composition unramified for every extension

Let $K$ be a number field and $S$ be a finite set of primes. Is it possible to construct a finite extension $M$ of $K$ such that $LM/M$ is unramified at (the primes above) $S$ for all degree $n$ ...
user246336's user avatar
  • 3,569
3 votes
1 answer
197 views

$H^1(\text{Gal}(L/K), O_L)=0$ for local fields

Let $K$ be a localfield and $O_K$be its ring of integers, and $L/K$ be a quadratic extension. It is known that $H^1(\text{Gal}(L/K), L) = 0$ according to Hilbert's Theorem 90. However, what is known ...
Poitou-Tate's user avatar
  • 6,353
5 votes
0 answers
72 views

Lang exercise 50 on Witt vectors

I'm reading the construction of the Witt ring from Lang's algebra. This is a series of exercises in chap. VI. In exercise 50 he says: If $x\in W_n(k)$ show that there exists $\xi\in W_n(\bar{k})$ ...
Mea Culpa's user avatar

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