Questions tagged [local-fields]
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262 questions
2
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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 $K$, ...
2
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0
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111
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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. ...
2
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0
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72
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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 ...
2
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0
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433
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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 $\psi_p$, ...
2
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0
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415
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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 ...
2
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0
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381
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Conductors of Weil-Deligne representations
Suppose $(V,N)$ is an $n$-dimensional semisimple $WD$ representation of $W_{\mathbb{Q}_p}$. This corresponds under local Langlands to an admissable representation $\pi$ of $GL_n(\mathbb{Q}_p)$. Is ...
1
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2
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268
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Is there any counterexample of the statement that the residue field extension of a separable extension is also separable?
Recently I'm reading Serre's Local Fields, and in page 22 (English version) he says that
The residue field extension $\overline L/\overline{K}$ is separable in each of the following cases (which ...
1
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3
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873
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Brauer group of complete DVR
Let $A$ be a complete discrete valuation ring with fraction field $K$ and perfect residue field $\kappa$.
Let $K_{nr}$ be the maximal unramified extension of $K$ and let $A_{nr}$ be its ring of ...
1
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1
answer
324
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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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1
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271
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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 ...
1
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1
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214
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Can two elements always belong to the same Laurent series field?
Let $\Omega$ be the completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the topology induced by the valuation $-\deg$ on $\mathbb F_q(T)$.
Let $x,y\in\overline{\...
1
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1
answer
89
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Compact subgroups of a linear group over non-Archimedean local field
$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers.
Is it true that any compact subgroup of $\GL_n(\mathbb{F})$ is conjugated to ...
1
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1
answer
141
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Existence of tamely ramified tower of extension over $\mathbb{Q}_p$
Let $p$ be a prime. There exist following containment : $$\mathbb{Q}_p \subset \mathbb{Q}_p^{\rm nr} \subset \mathbb{Q}_p^{\rm tr} \subset \overline{\mathbb{Q}}_p$$
Here $\mathbb{Q}_p^{\rm nr}$ and $\...
1
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1
answer
306
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Quadratic extension of local field
Let $F$ be a nonarchimedean local field of characteristic zero, and $E$ an extension of $F$ with $[E:F]=2^n$ for some $n$. Is it always possible to find a quadratic extension $M$ of $F$ such that $F\...
1
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1
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264
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How often does $-1$ have a square root in a local field?
Let $F$ be a nonarchimedean local field, say, charactersitic $0$. Is there any general theorem that tells when $\sqrt{-1}$ exists in $F$? How often does it happen?
1
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1
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284
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$K_v(a^{1/m}) /K_v$ is unramified if only if $v(a)≡0 \pmod m$
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 $v(a)≡0 \pmod m$.
This is from Silverman's ...
1
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1
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228
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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
https://math.stackexchange.com/q/1160086/187521
(I repost it here because I think it ...
1
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1
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163
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Is there any relationship between the study of class number of a number field with the study of class field theory through Lubin-Tate formal group?
I am curious to know if we can somehow relate to the study of local class field theory through Lubin-Tate formal group with the study of class number of a field (global field in general) in class ...
1
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1
answer
230
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Unramified composition 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 $n$ ...
1
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1
answer
190
views
Hilbert symbols vanishing
Let $p$ be an odd prime, and let $E/\mathbb{Q}_p$ be a finite extension that contains a primitive $p$-th root of unity $\zeta_p$ but not a primitve $p^2$-th root of unity $\zeta_{p^2}$. Let $a,b \in E^...
1
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1
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259
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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 ...
1
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1
answer
73
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Time complexity of Magma's `NormEquation` for quadratic extensions of $2$-adic fields
Note: This is similar to, but easier than, a previous question I asked here. It is a different question! I'm hoping this one might get an answer because it concerns a standard algorithm, whereas the ...
1
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1
answer
132
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Completion of $\mathbb F_q(T)$
It is easy to prove that for a an irreducible polynomial $P$ of degree $d$ of $\mathbb F_q[T]$, one can embed $\mathbb F_{q^d}$ in $\mathbb F_q(T)_P$ (the completion of $\mathbb F_q(T)$ at $P$) and ...
1
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1
answer
176
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Hilbert symbol over 2-adic field
Let $F$ be a finite extension of $\mathbb{Q}_2$, and let $(-,-)_F$ be the quadratic Hilbert symbol over $F$. Is the following true?
$(-1,-1)_F=1$ if and only if $\sqrt{-1}\in F$
1
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1
answer
138
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$0$-th Galois cohomology with topological Milnor K-groups coefficients
In local class field theory, the reciprocity map is constructed by using the isomorphism ${\rm Br}(F)\simeq \mathbb{Q/Z}$, where $F$ is a local field and ${\rm Br}(F)$ is its Brauer group. The ...
1
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1
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210
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isogenies between elliptic curves with multiplicative reduction
Let $ K $ be a $ p $-adic field. Suppose we have an isogeny of elliptic curves $ \phi : E \to E' $ defined over $ K $, where $ E $ and $ E' $ both have multiplicative reduction.
1) Is there anything ...
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1
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253
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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 $\...
1
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1
answer
216
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Factorisation of polynomials over finite field
Is there a method to factorise a polynomial, for $k \leq m$ and $a_i \in \mathbb{F}_p$,
$$
1 + t^k(1 + a_1 t + a_2 t + \ldots + a_m t^m)^k
$$
as a product $$
(1 + t^k)^{x_1} \cdots (1 + t^l)^{x_l} \...
1
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1
answer
173
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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 $K$...
1
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0
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25
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Characterising rank-$2$ lattices $\Lambda$ and conjugate-linear translate $g \sigma(\Lambda)$, given elementary divisors
Let $E/F$ be a quadratic unramified extension of local fields with $\operatorname{char} F = 0$. Let $\Lambda \subseteq E^2$ be an $O_E$-lattice of rank $2$. Let $g \sigma \in \operatorname{GL}_2(E)$ ...
1
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0
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124
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A question related to Kirillov model
I am reading Jackson - The theory of admissible representations of $\operatorname{GL}(2, F)$ and am not able to understand the following map related to Kirillov model. This result appears on p. 54:
I ...
1
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0
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180
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Maximal unramified extension and algebraic closure of $\operatorname{Frac}(\widehat{A_{\mathfrak{m}_A}})$
$\DeclareMathOperator\trdeg{trdeg} \DeclareMathOperator\Frac{Frac} $
Let $k$ be an algebraically closed field of characteristic $0$ and $K$ a function field over $k$. let $(A, \mathfrak{m}_A)$ a ...
1
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0
answers
88
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The localization map for the Mordell-Weil group of elliptic curves over finite Galois extensions
Let $L/K$ be a finite Galois extensions of number fields and $E/K$ be an elliptic curve. Denote by $\mathcal{F}$ the localization map
\begin{equation}
\mathcal{F}: H^1(G,E(L)) \rightarrow \bigoplus_{v ...
1
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0
answers
255
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Globalization of a local field
I am reading the paper ''Endoscopic classification of representations of quasi-split unitary groups'' by Chung Pang Mok, and cannot come up with the proof of theorem 7.2.1.
Here is the statement.
...
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0
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135
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What is the preimage of the maximal ideal under certain exponential functions?
I'm taking a shot in the dark with this question, so I apologize if it makes no sense.
Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $K_n$ be the field obtained by adjoining the $n$-th ...
1
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0
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132
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A question about Theorem 2.3.1 in Tate's thesis [closed]
I don't understand how to prove a conclusion in the Theorem.
When k is $p$-adic, the subgroups 1+$p^{v}$, $v>0$, of $u$ $(|u|=1)$ form a fundamental system of neighborhoods of $1$ in $u$, We must ...
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0
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169
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Why is $\overline{\mathbb{F}_p}((t))$ transcendental over $\mathbb{F}_p((t))$?
Why is $\overline{\mathbb{F}_p}((t))$ transcendental over $\mathbb{F}_p((t))$ ?
I guess $\overline{\mathbb{F}_p}((t))$ is not unramified over $\mathbb{F}_p((t))$ because $\overline{\mathbb{F}_p}((t))$ ...
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0
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164
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When is the extension $L(S)/L$ Galois and totally ramified?
Let $L$ be a finite extension of the $p$-adic field $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_K$ with uniformizer $\pi$. Let us consider the polynomial ring $L[x_1,x_2,\dotsc,x_l]$ in $l$-...
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0
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64
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Continous morphisms of a local field with conditions in positive characteristic
Let $P$ be a an irreducible polynomial of $k:=\mathbb F_q(T)$, $\Omega_P$ be the completion of an algebraic closure $\overline{k_P}$ of $k_P$, the completion of $k$ for the topology induced by the $P$-...
1
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0
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212
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Realization of a p-adic field as a completion of a number field
Let $F$ be a $p$-adic field of characteristic 0. Is it always possible to find a number field $K$ such that $K$ has only one place lying above $p$ and such that its completion at this place is $F$?
If ...
1
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0
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105
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Algebraic morphisms of affine varieties in positive characteristic
Let $\Omega$ be a completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the valuation $-\deg$.
Consider two matrices $M_1,M_2$ in $\mathcal M_2(\Omega)$ that are $\...
1
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0
answers
91
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Why does norm map the $\sigma$-conjugacy classes to the conjugacy classes?
Let $E/F$ be a cyclic extension of order $\ell$ (not assumed prime) of fields of characteristic $0$, and $\Sigma$ its Galois group; we denote by $\sigma$ a generator of $\Sigma$. We denote by $G(E), G(...
1
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0
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247
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compact $p$-adic Lie group can be embedded into $O_K^n$ or $\text{GL}_n(K)$?
Let $K$ be a local field of charecteristic $0$ and $G$ be a compact $p$-adic Lie group of dimension $n$, then can $G$ be embedded into $O_K^n$ or $\text{GL}_n(K)$ as a closed subgroup? This is a dual ...
1
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0
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108
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Characters of a quadratic extension and convergence
Let $F$ be a non-archimedean local field, $\chi$ a quasi-character of $F^\star$ and $\psi$ a positive character of $E^\star$. I would like to understand why the usual Rankin-Selberg zeta integrals ...
1
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0
answers
64
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Normgroup and the image of the Hilbert symbol are subgroups of index 2 in the principal units
Let $K$ be a local field over $\mathbb{Q}_2$ such that the extension $K(i)/K$ is ramified and let $U^1_{K(i)}$ and $U^1_K$ denote the groups of principal units in the fields $K(i)$ and $K$, ...
1
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0
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101
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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, \...
1
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0
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62
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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 $t_1,\...
0
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1
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346
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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}$ ...
0
votes
1
answer
146
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Does an affine building associated to a group satisfy the axioms of building?
Now I am reading Groupes réductifs sur un corps local : I. Données radicielles valuées written by Bruhat and Tits in 1972. Let $\Phi$ be a root system, $G$ a group with root data $(T,(U_{a},M_{a})_{a\...
0
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1
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211
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Some questions about splitting of sequence $0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$ for Henselian val field $K$
I have a couple of questions about following proof by Peter Scholze on splitting of the ses (...does it have a name?...)
$$0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$$
for $K$ henselian valuation ...