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Questions tagged [unramified-extension]

For questions related to unramified extension of local fields.

An extension $L/K$ of local fields is unramified if $[L : K] = [l : k]$ with $l = \mathcal O_L/π_L$ and $K = \mathcal O_K /π_K$ where $π_L, π_K$ are uniformizers of $L, K$. This is equivalent to saying that $π_K$ is inert in $L$, i.e. that the ramification index $e = v_K (π_L)$ is $1$.

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Tamely ramified extensions of $K_{\mathfrak p}^{\mathrm{unr}}$

I have some doubts regarding this statement, which I don't know if it's true: Statement: Let $K_{\mathfrak p}$ be the completion of a number field w.r.t. the $\mathfrak p$-adic valuation, and let $K_0$ be the maximal unramified extension of…
Marta Sánchez Pavón
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How to show that $(\sqrt{-17})$ is unramified in $\Bbb Q(\sqrt{-17}, \sqrt{17})$?

The Question: Let $K=\Bbb Q(\sqrt{-17})$ and $L=K(\sqrt{17})$. How to show that $\sqrt{-17}\mathcal O_K$ is unramified in $L$? Context: This is actually the last step I need to show that the Hilbert class field of $K=\Bbb Q(\sqrt{-17})$ is…
Milten
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Is the unramified local Langlands conjecture true?

I'm working on the unramified local Langlands conjecture and there is something that I don't understand if it is true or not. I want to start by saying that I don't care about endoscopic transfer or orbital integrals, but I just care about the…
Giulio Ricci
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Let $\Bbb{Q}_p⊆L_0⊆L$, where $L$ is extension of $\Bbb{Q}_p$ and $L_0$ be maximal unramified sub extension of $L/\Bbb{Q}_p$.

Let $\Bbb{Q}_p⊆L_0⊆L$, where $L$ is extension of $\Bbb{Q}_p$ and $L_0$ be maximal unramified sub extension of $L/\Bbb{Q}_p$. Let $O_0$ be ring of integers of $L_p$, $O$ be ring of integers of $L$. Then, why $O$ is finitely generated free module over…
Poitou-Tate
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Decomposition of unramified primes in normal closure of pure quintic field

Let $\Gamma$ = $\mathbb{Q}(\sqrt[5]{n})$ a pure quintic field, $k$ = $\mathbb{Q}(\zeta_5)$ the $5^{th}$ cyclotomic field, then $N$ = $\mathbb{Q}(\sqrt[5]{n}, \zeta_5)$ is the normal closure of $\Gamma$. Its known that a prime $p \in \mathbb{Z}$ such…
Fouad El
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Rings of integers for unramified extension $L_\mathfrak{q}/K_\mathfrak{p}$.

Let $L/K$ be an extension of number fields, $\mathfrak p\subseteq\mathcal O_K$ be a prime ideal, $\mathfrak{q}$ a prime lying above $\mathfrak p$. Here $L_\mathfrak q/K_\mathfrak p$ is a finite extension of non-Archimedean local fields with ring of…
Kango Mochizuki
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Base Change to turn Ramified Prime into Unramified

Let $L/K$ be finite extension of number fields, $O_K \to O_L$ of as schemes $\operatorname{Spec}(O_L) \to \operatorname{Spec}(O_L)$ the induced map of rings of integers. For $\frak{p} \ $$ \subset O_K$ prime ideal of $K$ one can talk about…
user267839
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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 unramified extension of degree $n$. We can denote by $F_{\infty}$…
Mario
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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 is true. And secondly, why is it so obvious that $L'/K$ is…
noradan
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Serre's Local Fields, Corollary 2 in Chapter 1.6. (unramified extension of local fields).

In Chapter I, Section 6 (Simple Extensions) of Serre's Local Fields, we see the following proposition and pair of corollaries: Assumptions: $A$ a discrete valuation ring, $B_f = A[X]/(f)$ for some polynomial $f$. $\mathfrak{m}$ is the unique maximal…
pedroelpanda
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Unramified quadratic extension of number field K is embedding into an unramified cyclic extension of degree 4 of K

is there any conditions over a number field K for an unramified quadratic extension of K to admit an embedding into an unramified cyclic extension of degree 4 of K?
ayoub-chess
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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 $\zeta_m$ in an algebraic closure of…
Shuster
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Maximal unramified extension inside a totally ramified local field extension

Let us consider the $p$-adic numbers $\mathbb Q_p$ and the consider the tower of Galois extensions: $$\mathbb{Q}_p \subset K \subset L; ~[L: \mathbb Q_p]<\infty$$ where $K$ is an unramified extension of $\mathbb{Q}_p$. $L$ is a totally ramified…
Learner
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The definition of the maximal tamely ramified extension.

Let $L_\mathfrak q/K_\mathfrak p$ be an extension of non-Archimedean local fields. Here $L/K$ is an extension of number fields. $L_\mathfrak q/K_\mathfrak p$ is tamely ramified if and only if $L_\mathfrak q/L_\mathfrak q^1$ is totally tamely…
Kango Mochizuki
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Base Change to resolve local Ramification

Let $L/K$ be finite extension of number fields with $O_K \to O_L$ the induced map of rings of integers. Let pick a prime $\frak{p}$ $ \subset O_K$ and a prime $\frak{P}$ $ \subset O_L$ lying over $\frak{p}$, and let complete $L/K$ with respect to…
user267839
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