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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use finite extensions approximate infinite extension

Let $K$ be a complete discrete valuation field, $K_\infty/K$ an infinite extension which is a directed union of finite extensions $\{K_i/K\}$. Let $L$ be a finite extension over $K$ and disjoint from $...
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Tamely ramified extension of local fields

I am trying to understand the tamely ramified extensions of a local field. First of all, when would be two such extensions $L_1/K$ and $L_2/K$ over the same base field be isomorphic? Second, for ...
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Describing Galois groups of some local fields

We can describe the Galois group of some global fields explicitly, for example, we can describe the Galois group of splitting field of $x^n-a$ over the rationals explicitly, especially the cyclotomic ...
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What does it mean for a field to be complete with respect to a topology?

This is from the Wikipedia page on Local Fields: "A field $K$ is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation $v$ and if is ...
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Rationality properties of unipotent groups defined over local fields of characteristic $p$

I know the following classical fact about rational points over a perfect, infinite field (p.s. everything is about linear algbaic groups, i.e. all groups are affine): If $G$ is a connected group ...
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injective continuous morphism of local fields

Let $q$ be a power of a prime $p$ and $P$ be an irreducible of $\mathbb F_q[T]$. The question is: Does there exist an injective continuous morphism between $\mathbb F_q((P))$ (with the $P$-adic ...
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Residue class field of $\mathbb{Q}_p(\zeta_m)$, where $(m,p) = 1$, is $\mathbb{F}_p(\zeta_m)$

I've been learning about local fields for some time now and I wanna prove the following: Let $p$ be a prime number and $m \in \mathbb{N}$ such that $(m,p) = 1$. Pick a primitive $m$-th root of unity $\...
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Understanding the Jacquet module of the Steinberg Representation

Let $G=GL_2(F)$ where $F$ is a non-Archimedean local field of characteristic $0$, for example $\mathbb{Q_p}$. Let $\chi=1_T$ be the trivial character of the maximal split torus $T=\begin{pmatrix}* &...
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Ramification of the field extension $\mathbb{Q}_p(\zeta_{p^2},p^{1/p})$

I am interested in the extension of local fields $\mathbb{Q}_p(\zeta_{p^2},p^{1/p})/\mathbb{Q}_p$. Is it totally ramified? Here are the partial results that I have: It is a field extension of degree $...
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Is the "order of the decomposition group of an infinite prime" equal to $1$? $|D_{\mathfrak{P}}|=1$? (About infinite primes and decomposition groups)

Sorry for this very easy question. I am reading Janusz's book "Algebraic Number Fields". I am in section 2 of the fifth chapter of Janusz's book "Algebraic Number Fields", exactly ...
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Does $\mathbb{Q}_2(\sqrt{-1},\sqrt{10})$ contain $\mathbb{Q}_2(\sqrt{5})$?

My suspicion is yes, because I think $\mathbb{Q}_2(\sqrt{-1},\sqrt{10})$ is not totally ramified over $\mathbb{Q}_2$, so it should contain the unique unramified extension, $\mathbb{Q}_2(\sqrt{5})$. If ...
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1 answer
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How to do integration of matrices on local fields?

I'm currently reading Buzzard's note and trying to calculate the integral on page 5: $$ S(f)(\mathrm{diag}(\varpi, 1)) = q^{-1/2} \int_N f \left( \begin{pmatrix} \varpi & \varpi n \\ 0 & 1 \...
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How to distinguish quadratic ramified extensions of $\mathbb Q_p$?

Let $p \not = 2$ be an odd prime number. I am trying to understand how to characterize isomorphism classes of quadratic ramified extensions of $\mathbb Q_p$. Any such extension $E$ is isomorphic to a ...
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5 votes
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Parabolic induction of p-adic groups independent of the choice of parabolic.

I noticed many papers concerning the theory of smooth representations of connected reductive p-adic groups, omit the mention of the specific parabolic subgroup $P\subseteq G$ used in defining the ...
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For a number field $K$, the ring of integers of $K \otimes_{\mathbb{Q}} \mathbb{Q}_p$ is $\mathcal{O}_K \otimes_\mathbb{Z} \mathbb{Z}_p$

Let $K$ be a number field and let $p$ be an odd prime. There is a natural map $$ \mathcal{O}_K \otimes_\mathbb{Z}\mathbb{Z}_p \to K \otimes_\mathbb{Q}\mathbb{Q}_p,\quad \alpha \otimes n \mapsto \alpha ...
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Can $\Bbb{Z}$ be regarded as a $\Bbb {Z}_p$-module? [closed]

Can $\Bbb{Z}$ be regarded as a $\Bbb {Z}_p$-module, where $\mathbb{Z}_p$ is the set of $p$-adic integers? I know $\Bbb{Z}$ cannot be $\Bbb {Z}_p$-algebra, so I cannot make $\Bbb{Z}$ into $\Bbb {Z}_p$...
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example of $L/\Bbb{Q}_p$ such that there is no prime element $π$ of ring of integers $L$ such that $p=π^e$.

Let $L/\Bbb{Q}_p$ be ramification index $e$ extension. Let $π$ be prime element of $L$. Then, $p=π^eu$ ($u$: unit element of ring of integers of $L$)。 But I have met the remark that we can choose ...
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Let $L/\Bbb{Q}_p$ be ramification index $e$ extension. Let $π$ be prime element of $L$. Then, why $p=π^eu$(u:unit element of ring of integers of $L$)?

Let $L/\Bbb{Q}_p$ be ramification index $e$ extension. Let $π$ be prime element of $L$. Then, why $p=π^eu$ ($u$: unit element of ring of integers of $L$)? From the definition of ramification index, $(...
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Eisenstein criterion for Local fields $\mathbb{F}_q((t))$ [closed]

Let $t$ be an indeterminate, and $\mathbb{F}_q(t)$ be the field of rational functions over $\mathbb{F}_q$ ($\mathbb{F}_q(t)$ is a global field). Denote by $\mathbb{F}_q((t))$, the completion of $\...
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2 votes
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Question on integration on a local field

Let $F$ be a non-Archimedean local field, and $\mu_F$ a Haar measure on $F$. The space $C^{\infty}_c(F)$ of locally constant functions of compact support is spanned by characteristic functions of the ...
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Local Fields book with lots of exercises [duplicate]

I really like Marcus's 'Number Fields' because it has so many exercises. I'm trying to find a similar book about Local Fields, but all the books I've looked at (Neukirch, Cassels, Guillot, Serre) have ...
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1 answer
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Ramified extension of local field which is not Galois

Unramified extension of local field is automatically galois because there is bijection between unramified extension of local field and extension of residue finite field, that is galois. But, what ...
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Lemma 8.1 of "The Local Langlands Conjecture for GL(2)"

Let $F$ be a non-Archimedean local field, and $N$ the subgroup of $G=GL_2(F)$ of the form $\begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}$ with $x \in F$. Let $(\pi,V)$ be a smooth ...
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How does ramification groups help to study ramifications of local field extension?

I am studying higher ramification groups of local field extension. In Wikipedia it is mentioned that higher ramification groups gives information about ramification of extension. Suppose $L/K$ be a ...
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Do ramification groups contain non-central abelian normal subgroups?

I am studying the proof of integrality of the conductors of Galois representations from these notes, and I have hit a roadblock in a step of the proof of Proposition 3.1.40 (page 57). The setting is ...
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Explicit Local Fundamental Class

Let $L/K$ be a Galois extension of local fields of degree $n:=[L:K]<\infty$ with Galois group $G:=\operatorname{Gal}(L/K)$. In short, my question is as follows. Is there an explicit computation of ...
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16 votes
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Is the power of $2$ in the Euclidean norm related to the fact that the algebraic closure of the reals is $2$-dimensional?

Consider any local field $K$, endowed with its topological field structure. We define the function $| \cdot | : K \to \mathbb{R}_{\ge 0}$ as $$|x| = \frac{\mu(xS)}{\mu(S)},$$ where $\mu$ is any Haar ...
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1 vote
1 answer
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Galois action on a uniformizer of a non-archimedean local field

Suppose that $k$ is a non-archimedean local field. Let $K/k$ be a tamely, totally ramified degree-$3$ extension of $k$. If $\varpi$ is a uniformizer of $K$, I want to understand the relationship ...
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2 votes
1 answer
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Convergence of a sequence in a non-Archimdean valuation for a field.

Admittedly, this is a homework question and so please just hint me where to go with this question if it is ok, thank you! So suppose $k$ is a field and $|.|$ is a non-Archimedean valuation. In other ...
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2 answers
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Definition of $\mathfrak{p}$-adic field in Cassels

I am trying to understand the definition of a $\mathfrak{p}$-adic field given in Cassels' Local Fields (Cambridge University Press, 1986, page 144). Here is what he says: Definition 1.1. Let the ...
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4 votes
1 answer
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An infinite family of Artin-Schreier polynomials which all split in $\mathbf{F}_q(\!(\theta)\!)$

Let $\mathbf{F}_q$ be a finite field with $q$ elements and let $K$ denote the local function field $\mathbf{F}_q(\!(\theta)\!)$. Let $R$ be its valuation ring $\mathbf{F}_q[\![\theta]\!]$. Let $u$ be ...
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3 votes
2 answers
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Calculating the ramification and inertia degree of $\mathbb{Q}_2(\sqrt{3}, \sqrt{7})$ and $\mathbb{Q}_2(\sqrt{3}, \sqrt{2})$ over $\mathbb{Q}_2$

My question 1 is how to calculate the ramification index and inertia degree of $K_1 := \mathbb{Q}_2(\sqrt{3}, \sqrt{7})$ and $K_2 := \mathbb{Q}_2(\sqrt{3}, \sqrt{2})$ over $\mathbb{Q}_2$. My attempts ...
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3 votes
3 answers
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Are rings of power series over a local field complete?

Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $D$ be some disk $D = \{ x\in \overline{K} \mid |x| < c < 1\}$. Is the set of power series in $K[[T]]$ which converge on $D$ and are ...
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Using Hensel lemma to find roots in valuation rings

Let $K$ be a p-adic field with absolute Galois group $\varGamma$. Let $O^{ur}_K$ be the ring of integers of $K^{ur}$. Then, for $n$ prime to $p$, the $\varGamma$-module $\mu_{n}$ is unramified. Hensel ...
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Let $K$ be a local field and $K'$ be it's finite extension. Then, why ${K'}^{nr}/K^{nr}$ is finite extension?

Let $K$ be a local field and $K'$ be it's finite extension. Then, why ${K'}^{nr}/K^{nr}$ is finite extension ? Unramified extension of local field corresponds to extension of residue field. So, I can ...
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1 answer
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Definition of discriminant of local field

Let $K$ be a local field, and $L$ be an finite extension of $K$. Then, $L$ is also local field. Then, what is the definition of discriminant of extension L/K ? Discriminant of extension of $\mathbb Q$ ...
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2 answers
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Inertia group $I_K$ is isomorphic to $Gal(\bar K /K^{nr})$ as a group

Let $K$ be a local field. Then, elements of $Gal(\bar K /K)$ which induces identity map on residue field forms group, and the group is called inertia group of $Gal(\bar K /K)$, and written like $I_K$....
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type of reduction does not change under unramified extension

Let $K$ be a local field, and $E/K$ be an elliptic curve over $K$. Let $K'/K$ be unramified extension, then , the reduction type of $E/K'$ is the same as $E/K$. My proof. Let $E$ be minimal ...
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In what kind of situation, ring of integers of $\mathbb{Q}_p(α) $ is $\mathbb{Z}_p[α] $?

In what kind of situation, ring of integers of $\mathbb{Q}_p(α) $ is $\mathbb{Z}_p[α] $? For example, if $α$ is primitive root of unity, then it holds. But it does not hold in general, for example, ...
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Ring of integers of $\mathbb{Q}_p(α) $ [duplicate]

Is ring of integers of $\mathbb{Q}_p(α) $ is $\mathbb{Z}_p[α] $ ? If $α$ is primitive root of unity, dependentable reference reads that the titled statement holds. But in general, does the statement ...
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Definition of inertia group $I_v$

Inertia group of $Gal(\bar K/K)$ is defined as set of elements of $Gal(\bar K/K)$ that act trivially on the residue field $\bar k$. We often denote inertia group of $Gal(\bar K/K)$ by $I_v$. But ...
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Example of unramified set at $v$

Let $Σ$ be a set on which $Gal(\overline{K}/K)$ acts. We say that $Σ$ is unratified at $v$ if the action on $I_v$(the set of elements of $Gal(\overline{K}/K)$. For example, let $E$ be elliptic curve ...
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1 vote
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$j∈R$ and $256(1-λ(1-λ))^3-j λ^2(1-λ)^2=0$ implies $λ∈R$ and $λ$ is not congruent to $0$ or $1$ mod $M$

Let $K$ be a local field whose residue field's is not characteristic $2$, and $R$ be it's ring of integers, and $M$ be $R$'s maximal ideal. Let $j∈R$, then, I would like to prove $256(1-λ(1-λ))^3-j λ^...
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1 vote
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Question about p.165 in Local Fields by Cassels

I am trying to understand the assertions made at the end of the paragraph below. I am not sure if I am missing some certain facts from previous chapters, but I do not see the claims made in bold ...
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Example of degree $n$ ramified, but not totally ramified extension

I'm looking for an example of degree $n$ ramified but not totally ramified example over $\Bbb Q_p$. I can find degree $n$ ramified extension, for example, $\Bbb Q_p({p^{1/n}})/\Bbb Q_p$. $p^{1/n}$'s ...
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Let $K$ and $K'$ be local field, and $v$ and $v'$ be it's valuation. Suppose $K'/K$ is unramified

Let $K$ and $K'$ be local field, and $v$ and $v'$ be it's valuation. Suppose $K'/K$ is unramified, then, for all $u'∈K'$,there exists $u∈K$ such that $u/u'∈R'^×$. My attempt: To probe this, $v'(u/u')=...
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0 votes
1 answer
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Sufficient condition of $K'/K$ is ramified extension

Let $K$ be a local field and $K'$ be it's finite extension. And there exists $a∈K'$ such that $v(a)$ is not integer. Then, ramification index $e$ of $K'/K$ is at least 2, in other words, $K'/K$ is ...
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Completions of Global Field are Local Fields

Neukirch's book defines a global field as a finite extension of $\mathbb Q$ or a finite extension of $\mathbb{F}_p(x)$ for $p$ prime, and a local field as a field that admits some discrete valuation $...
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Index $n$ subgroup of local field exists?

Let $K$ be a local field. Let $n$ be arbitrary positive integer. Does $K^×$ have index $n$ subgroup? If I could prove this, from local class field theory, I can say $K$ has arbitrary degree of ...
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$K_v(a^{1/m}) /K_v$ is unramified if only if $v(a)\equiv0 \pmod m$

Let $K$ be a number field and $v$ be it's one of $K$'s non-archimedian valuation. Then, I would like to prove $K_v(a^{1/m}) /K_v$ is unramified if only if $v(a)≡0 \pmod m$. I know unramified ...
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