# 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 $[\pi]$ ...
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1 vote
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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 ...
1 vote
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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 $p$. ...
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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
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### 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 ...
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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. ...
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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 $q$ ...
• 403
1 vote
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### 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 ...
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### 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 ...
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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&...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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1 vote
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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 ...
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### 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 $\hat{\mathfrak{p}}$ ...
• 1,016
1 vote
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### 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 ...
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1 vote
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### 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 $n$ ...
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### $H^1(\text{Gal}(L/K), O_L)=0$ 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 ...
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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 $\xi\in W_n(\bar{k})$ ...