All Questions
9 questions with no upvoted or accepted answers
10
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0
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1k
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Automorphisms of local fields
It is an amusing coincidence (at least it appears to be a coincidence to me) that any completion of the field $\mathbb{Q}$ has trivial automorphism group as an abstract field, i.e. when ignoring the ...
- 1,298
6
votes
0
answers
150
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$SL_2(\mathbb{Z}_p)$ extension of a local field
Let $G$ be an arbitrary open group of $SL_2(\mathbb{Z}_p)$ and $K$ be a finite extension of $\mathbb{Q}_p$. Can we construct a Galois extension field $E$ of $K$ such that $\text{Gal}(E/K)\cong G$? ...
user141691
5
votes
0
answers
252
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Explicit construction of abelian wild inertial extensions of maximal tamely ramified extension of $\mathbb{Q}_p$?
In Iwasawa's paper On Galois groups of local fields, he proves that if $V$ is the maximal tamely ramified extension of $\mathbb{Q}_p$, with Galois group $\Gamma$ over the base, then its abelianized ...
- 2,054
3
votes
0
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348
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Question about Corollary 7.3 from Silverman's The Arithmetic of Elliptic Curves
I am trying to understand an argument of Corollary 7.3 from Silverman's The Arithmetic of Elliptic Curves. I am stuck and I would appreciate any explanations.
Let $E$ be an elliptic curve over $K$, ...
- 3,625
2
votes
0
answers
132
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For a quadratic extension $E/K$, condition on a character $\chi:E^\times/E^{\times 2} \to C_2$ to give a $C_4$-extension $L/K$
Let $K$ be a finite extension of $\mathbb{Q}_2$, and let $E/K$ be a quadratic extension. By local class field theory, quadratic extensions $L/E$ correspond to quadratic characters $\chi:E^\times \to ...
- 411
2
votes
0
answers
182
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Reference request: Regarding the image of inertia group being a subgroup of Aut($\widetilde{E}$)
Let $E$ be an elliptic curve over $\mathbb{Q}_p$ with potential good reduction. I was told that if $F$ is the smallest Galois extension over $\mathbb{Q}_p$ such that $E$ has good reduction then the ...
- 3,625
1
vote
0
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165
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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 $l$-...
- 930
0
votes
0
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93
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Existence of maximal totally ramified subextension
Suppose $K/\mathbb Q_p$ is a finite extension, with ramification index $e$ and inertia index $f$. I want to ask whether there exists a subextension $L/\mathbb Q_p$ totally ramified of degree $e$?
- 785
0
votes
0
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78
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Local field such that the value group of $K^\text{perf}$ ( perfect closure of $K$) is $\bigcup_{n\geqq1}(1/p^n) \Bbb{Z}$
Let $K$ be a local field of positive characteristic.
I'm looking for a $K$ which satisfies the following condition.
Value group of $K^\text{perf}$ (perfect closure of $K$) is $\bigcup_{n\geqq1}(1/p^n)...
- 1,541