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.
484
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Geometric interpretation of Lefschetz number for local fields
I have the following problem.
Let $L/K$ be a finite galois extension of local fields with Galois group $G$. For nontrivial $g\in G$ define Lefschetz number $i_{L/K}(g):= \min\limits_{x\in \mathcal{O}...
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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 $\xi\in W_n(\bar{k})$ ...
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Factorization of polynomials in $\mathbb{Q_p}[X]$
I stumbled upon these two questions while reading Milne's notes on Algebraic Number Theory.
Milne's problem 7-6: Let $\gamma=\sqrt{p_1}+\cdots+\sqrt{p_n}$, $p_i$ are distinct primes. We could prove ...
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34
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Absolute values and Galois action
Let L be a finite Galois extension of the local field K.Then why is the following true?
$$|\sigma(x)|=|x|,\forall \sigma\in\text{Gal}(L/K),x\in L$$
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What is the local character given by class field theory?
Dorman's paper on singular moduli uses the "local character given by class field theory". Specifically, on page 178, the author states,
For each prime $p$, finite or infinite, let $\...
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Formal groups over discrete valuation rings
Let $k/\mathbb{Q}_p$ be a finite extension. Let $k^{ur}$ be its maximal unratified extension and consider $K$ its completion. Let $\varphi$ be the Frobenius morphism in Gal$(k^{ur}/k)$ and whit the ...
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48
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Absolute Galois group of the p-adic completion of a valuation field
Let $K$ over $\mathbf{Q}_p$ be an algebraic extension, not complete for $|\cdot|_K$ (the unique extension of $p$-adic norm $|\cdot|_p$ to $K$).
(For example, $K=\cup_n\mathbf{Q}_p(\zeta_{p^n})$ the ...
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A localization sequence for étale cohomology of $\mathcal O_K$, where $K$ is a local field
Given a non-archimedean local field $K$, let $\mathcal O_K$ be the associated valuation ring and $k$ its residue field. According to this MO answer, we have short exact sequence
$$0 \to H^2(\mathcal ...
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Equivalence class of valuations on a field.
I have been reading Cassels Local Fields and I have a couple of questions regarding valuations that satisfy the triangle inequality.
Cassels defines a valuation on a field $k$ as $|.|:k\to \mathbb{R}$ ...
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Etale cohomology of $Spec(𝔽ₚ^{\text{sep}}((t)))$
I am thinking about how norms $ν : L ⭢ ℤ$ on higher local fields could induce long exact sequences in different cohomologies.
$𝔽ₚ^{\text{sep}}((t)),ℚₚ^{\text{sep}}$, and $ℂ$ are a local fields. What ...
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Local fields vs. self-Pontryagin-dual locally compact fields with self-dual Schwartz-Bruhat functional
The local fields $\mathbb{R}$, $\mathbb{C}$, $\mathbb{Q}_p$, and the Adele ring $\mathbb{A}$ are all Pontrjagin dual to themselves (self duality). The consideration of the multiplicative Fourier ...
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Definition of a non-split root subgroup
I've been reading through Tits's Corvallis survey "Reductive groups over a local field" and something that surprised me that was taken for granted about the definition of root subgroups. Let ...
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Trivial Artin map on $F(\zeta)_{\frak P}$
It's exercise 6.4 in N. Childress - Class F. Th.
$q$ is a prime number $F\subset L$ an cyclic extension $[L:F]=q$; $\zeta$ a $q$-root of 1.
One have shown ${\cal A}_{L/F}({\bf i}(F_{{\frak p}}))=1$ ...
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More examples of non-split algebraic groups
I'm reading Reductive Groups over Local Fields by Tits (from the Corvallis proceedings), and I'm having trouble making sense of several of the definitions, especially when it comes to the local Dynkin ...
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Explicit calculation of norm groups in Q_p
Let $p$ be a prime and $(p, n)=1$. I am wondering how to explicitly compute the norm groups of $\mathbb{Q}_p[\zeta_n]/\mathbb{Q}_p$. Ideally, I would like a computation using class field theory and ...
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Showing degree of global extension is sum of degree of local extensions
I am trying to understand why, for a global extension $F \subset K$, given a place $\mu$ over $F$, we have that $$n = \sum_{\nu: \nu | \mu} n_{\nu}$$
where $n = [K:F]$, $n_{\nu} = [K_{\nu}|F_{\mu}]$ ...
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Is a field with archimedean absolute value with compact unit sphere complete?
By the unit sphere of a valued field $(K,|\cdot|)$ I mean $\{x\in K:|x|=1\}$.
We know that if the unit closed ball $\{x\in K:|x|\le 1\}$ of a valued field $K$ with nontrivial absolute value is compact,...
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When is a local extension unramified by discriminant
This is a question from the proof of Proposition VIII.1.6 in Silverman's Arithmetic of Elliptic Curves. We have a number field $K$ and a place $v$; $\operatorname{ord}_v$ is the normalized valuation ...
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2
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251
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Number of extensions of a local field of fixed degree
I'm trying to understand the proof of the following theorem.
A local field $K$ of characteristic $0$ admits (inside a fixed algebraic closure) only finitely many extensions of a given degree.
The ...
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2
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Is $v((f \circ g)(x)) \geq v(f(x))+v(g(x))$, where $v$ is p-adic valuation?
Let $\mathbb Q_p$ be a $p$-adic field and $f,g \in \mathbb Q_p[x]$. Suppose $v$ is a $p$-adic valuation, and let $f \circ g$ be the formal composition.
Is $v(f \circ g) \geq v(f)+v(g)$ ?
For the ...
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Conditions for the tower of Galois extensions of local fields to be a Galois extension
I have the following question.
Let $L/K$ be cyclic extension of local fields, $\sigma$ is a generator of $\text{Gal}(L/K)$, and $M/L$ is a finite abelian extension. Prove that:
$M/K$ is a Galois ...
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Classifying abelian extensions of number fields with class field theory
I am reading about local class field theory here https://math.mit.edu/classes/18.785/2015fa/LectureNotes24.pdf and in the end the author mentions an application to count the number of Galois ...
- 471
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Why $[E(K_q) : E_0(K_q)] ≤ 4$ and $E_0(K_q)$ is divisible by $2$ implies $E(K_q)[2^∞] = E[2]$?
Let $E$ be an elliptic curve over $\Bbb{Q}$ with no non-trivial rational $2$-torsion point.
We write $q$ for a odd prime of quadratic number field $K$. Let $K_q$ denote the $q$-adic completion of $K$. ...
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Norm residue map maps uniformizing elements to Frobenius elements
I am reading Algebraic Number Theory by Cassel, Frohlich. I have a question about the proof that norm residue map maps uniformizing elements to Frobenius elements in unramified extension. This is in ...
- 471
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Cohomology of local fields in positive characteristic
It is well-known from local class field theory that the Brauer group $\text{Br}(k)$ of a local field $k$ gets killed as you pass to sufficiently large extensions of $k$. In particular, $\text{Br}(L)(p)...
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Discriminant of a Quaternion Algebra over a local field
I am reading Voight's "Quaternion Algebras" and I have a problem
with Example 29.7.6. which is about the discriminant of a Quaternion Algebra $B$ over a local field $F/K$.
The equation is
$$...
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On the concept of primary element
Let $\ell$ be an odd prime number,$\zeta_{\ell}:=e^{2\pi i/\ell}$, $F:=\mathbb{Q}(\zeta_{\ell})$ be a cyclotomic field, $\mathcal{O}_F$ be its integer ring and $\lambda:=1-\zeta_{\ell}$.
[Ireland-...
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What is the decomposition of global units $1+\mathfrak{p}$?
Let $p \geq 2$ be prime and $K=\mathbb Q(\zeta_p),~\zeta^{p}=1$ with ring of integers $\mathcal{O}_K$. we denote by $\mathfrak{p} \mid p$ the prime ideal of $K$ dividing $p$. Let $K_{\mathfrak{p}}$ be ...
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Tate cohomology of units
Show that for any local field K, there is a finite Galois extension L, such that $H^{i}_{T}(Gal(L/K),O_L^*)$ does not vanish for all i, here $O_L$ is ring of algebraic integers of L.
I only know that ...
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uniqueness of local Artin map
This problem defines a map with the same properties as local Artin map and asks you to prove they are equal. I'm having problems with b) and c). Is the first part of b) comes from Galois ...
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Why's $v_L (x)=\frac{1}{n}v_K(N_{L/K}(x))$ integer for unramified local field extension $L/K$?
If $K$ is a local field and $L/K$ is a finite extension, then the valuation $v_K$ can be extended uniquely to a valuation $v_L$ of $L$ such that $v_L$ restricted to $K$ is equal to $v_K$. This is one ...
- 5,567
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What is the quotient group $\mathfrak{q}^2/\mathfrak{p}^2\mathbb Z_p$?
Let $p \geq 2$ be prime and $K=\mathbb Q(\zeta_p),~\zeta^{p}=1$ with ring of integers $\mathcal{O}_K$. we denote by $\mathfrak{p} \mid p$ the prime ideal of $K$ dividing $p$. Let $K_{\mathfrak{p}}$ be ...
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Discriminant of a $V_4$-extension of local fields is the product of discriminants of intermediate fields
Let $L/K$ be a Galois extension of $p$-adic fields with Galois group $V_4 = C_2 \times C_2$, and write $d_{L/K}$ for its discriminant, which is an ideal of $\mathcal{O}_K$. The extension $L/K$ has ...
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Splitting on $p$ adic unit group
Let $p$ be a prime.
It is basic that following isomorphisms;
$\mathbb{Z}_p^{\times}/(1+p^n\mathbb{Z}_p )\cong (\mathbb{Z}_p/p^n\mathbb{Z}_p)^{\times} $,
$(1+p^n\mathbb{Z}_p)/(1+p^{n+1}\mathbb{Z}_p )\...
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Tate Twists of Z/nZ
Let $k$ be a $p$-adic local field with absolute Galois group $G_k$. In Cohomology of Number Fields, the authors define the $n$-th Tate twist of a finite $G_k$-module $A$ as the $G_k$-module $A(n)$ ...
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Basic proofs on Weil groups
I had some questions regarding the Weil group. We defined a surjective map $res: \operatorname{Gal}(L/K) \rightarrow \operatorname{Gal}(k_L/k)$, where $k_L,k$ is the residue field of $L,K$ ...
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Is the subgroup of $WC(E/K)$ of curves with an $L$-rational point finite for $K$ local, $L/K$ finite Galois?
User BrauerManinobstruction's question "Weil–Châtelet group of a real elliptic curve is isomorphic to
Z/2Z when Δ>0" relates to exercise 10.7 of Silverman's AEC, which asks the reader to ...
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Step in Yoshida's proof of Hasse-Arf theorem
This question concerns Yoshida's proof of the Hasse-Arf theorem in the local class field theory in https://arxiv.org/abs/math/0606108 (page 16).
For a totally ramified extension $K′/K$ of local fields,...
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If there exists a prime $l$ such that $l$ spilts in $K$, why does that imply there exists a place of $K$, which satisfies $K_v \cong \Bbb{Q}_l$?
Let $K$ be a number field.
If there exists a prime $l$ such that $l$ spilts in $K$, why does that imply there exists a place of $K$, which satisfies $K_v \cong \Bbb{Q}_l$ ?
Here, $K_v$ denotes ...
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Surjectivity of norm map on principal units in a totally ramified extension of local fields
I have a question about a statement in Chapter V, section 3 of Serre’s book on local fields.
Let $\ell$ be a prime number, and let $L/K$ be a cyclic Galois extension of local fields of degree $\ell$, ...
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When is a norm of a formal power series over a local field a polynomial?
Let $K$ be a finite extension of $\mathbb Q_p$ and $L/K$ be a finite Galois extension. Then also $L(T)/K(T)$ and $L((T))/K((T))$ are Galois extensions with Galois group isomorphic to ${\rm Gal}(L/K)$ ...
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Why Chebotarev's density theorem implies $ \exists {v \in Ω_L}$ such that $Gal(L_v/K_{v'})$ is generated by $\sigma^2$?
Let $L/K$ be an extension of number field.
Let $Gal(L/K)$ is generated by $\sigma$.
Why Chebotarev's density theorem implies $ \exists {v \in Ω_L}$ such
that $Gal(L_v/K_{v'})$ is generated by $\sigma^...
- 5,567
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Example of rational jumps in upper ramification filtration
Can anyone point to me or write down an explicit example of a non-integer jump of the higher ramification filtration in positive characteristic and the corresponding equations of the intermediate ...
- 1,115
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Do formal Laurent series of norm 1 over a local field come from power series over ring of integers?
Let $K/\mathbb Q_p$ and $L/K$ be finite extensions of fields, with $O_K$ resp. $O_L$ being the rings of integers in $K$ resp. $L$. Let $N:L\to K$ denote the norm map from $L$ to $K$. Consider the ...
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44
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A uniformizer of a finite extension of $\mathbb{Q}_p$
Let $L$ be a finite extension of $\mathbb{Q}_p$, say $[L:\mathbb{Q}_p]=n$. The uniformizer of $\mathbb{Q}_p$ is $p$, let we assume that $\sqrt[n]{p} \notin L$ and consider $K=L(\sqrt[n]{p})$. Is $\...
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Finding $\dfrac{\sigma(u)}{u}=\alpha$ for local field $K$, $L$ unramified over $K$, $\sigma$ being Frobenius of $L/K$ and $\alpha$ having norm 1.
Let $K$ be a nonarchimedean local field, $L$ be the unique degree $n$ unramified extension of $K$. Let $\sigma$ be the Frobenius element of $L/K$, that is, $\sigma\in\mathrm{Gal}(L/K)$ such that $\...
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1
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JS Milne Algebraic Number Theory 8.6 : normalized absolute values for local fields
This is from JS Milne's notes on Algebraic Number Theory, lemma 8.6
Let $K$ be a local field with normalized absolute value $|.|_K$. Let $L$ be a finite separable extension of degree $n$, and $|.|_L$ ...
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29
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Notation in the Tate-Nakayama Theorem
In ch. IX, §8 of Serre’s Local Fields, we find the Tate–Nakayama theorem, an essential lemma for class field theory:
Theorem 14. Let $G$ be a finite group, $A$ a $G$-module, and $a\in H^2(G,A)$. Let $...
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Elements of a local field with trace in $\mathbb Z_p$
Let $K_p$ be a finite extension of $\mathbb Q_p$. Then we have the trace map:
$$
T:=\operatorname{Tr}_{K_p|\mathbb Q_p}:K_p\to\mathbb Q_p
$$
Is there any characterization of the open set $T^{-1}(\...
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Norm of a root of unity $\zeta$ is $1$ only if $\zeta=1$?
Let $p$ be an odd prime number. Let $K/\mathbb Q_p$ be a finite extension, with $K$ having residue field $\mathbb F_q$ of order $q$ some power of $p$. Let $\zeta\in\mu_{q-1}\subset K$ be a $(q-1)$st ...