Unanswered 'local-fields+galois-theory' Questions

10 votes

0 answers

1k views

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 ...

Robert Kucharczyk

1,298

asked Nov 12, 2011 at 13:31

8 votes

0 answers

317 views

Finding a cyclic cubic extension of a field

Let $K$ be a field and let $E/K$ be a Galois extension of degree 6 with $\text{Gal}(E/K) = S_3$, the symmetric group on 3 letters. Pick two different transpositions $s_1, s_2$ in $S_3$ (hence $s_1s_2$ ...

thierry stulemeijer

1,017

asked Apr 20, 2018 at 16:36

6 votes

0 answers

513 views

Extensions of p-adic number fields

Let $p$ be a prime number and $\mathbb{Q}_p$ be the $p-$adic rational field. Let $E/\mathbb{Q}_p$ be a fixed finite extension. On this site, I define a finite extension $F/E$ to be "good" if ...

Eric

71

asked Nov 12, 2023 at 14:45

6 votes

0 answers

150 views

$SL_2(\mathbb{Z}_p)$ 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

asked Jun 4, 2020 at 6:29

5 votes

0 answers

251 views

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 ...

xir

2,054

asked Oct 6, 2019 at 22:59

3 votes

0 answers

128 views

Galois cohomology with coefficients in the integers of the Lubin-Tate extension

Let $K$ be a $p$-adic local field, and $L$ the Lubin-Tate extension obtained from $K$ by attaching roots of some Lubin-Tate formal $\mathcal{O}_{K}$-module with $Gal(L/K) \simeq \mathcal{O}_{K}^{\...

Piotr Pstrągowski

2,197

asked Oct 3, 2022 at 12:58

3 votes

0 answers

135 views

Conjugation action of $Gal(\bar{s}/s)$ on the tame ramification group

There is a statement in SGA 7-1 Exposé 1 (P. Deligne, Résumé des premiers exposés de A. Grothendieck, pdf of SGA7-1), (0.3.1): $S$ is a Henselian trait (i.e. the spectrum of a henselian discrete ...

user545

83

asked Dec 10, 2019 at 20:32

3 votes

0 answers

347 views

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 $K$, ...

Johnny T.

3,625

asked Jan 28, 2018 at 13:27

2 votes

0 answers

132 views

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 ...

Sebastian Monnet

411

asked Nov 13, 2023 at 13:19

2 votes

0 answers

181 views

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 ...

Johnny T.

3,625

asked Jan 27, 2018 at 20:02

1 vote

0 answers

164 views

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 $l$-...

MAS

930

asked Aug 20, 2021 at 11:41

1 vote

0 answers

91 views

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(...

M masa

479

asked Jun 29, 2020 at 13:02

0 votes

0 answers

93 views

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 $e$?

Richard

785

asked Jun 29 at 7:49

0 votes

0 answers

78 views

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)...

Duality

1,541

asked May 23, 2022 at 14:37

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All Questions

Automorphisms of local fields

Finding a cyclic cubic extension of a field

Extensions of p-adic number fields

$SL_2(\mathbb{Z}_p)$ extension of a local field

Explicit construction of abelian wild inertial extensions of maximal tamely ramified extension of $\mathbb{Q}_p$?

Galois cohomology with coefficients in the integers of the Lubin-Tate extension

Conjugation action of $Gal(\bar{s}/s)$ on the tame ramification group

Question about Corollary 7.3 from Silverman's The Arithmetic of Elliptic Curves

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 $L/K$

Reference request: Regarding the image of inertia group being a subgroup of Aut($\widetilde{E}$)

When is the extension $L(S)/L$ Galois and totally ramified?

Why does norm map the $\sigma$-conjugacy classes to the conjugacy classes?

Existence of maximal totally ramified subextension

Local field such that the value group of $K^\text{perf}$ ( perfect closure of $K$) is $\bigcup_{n\geqq1}(1/p^n) \Bbb{Z}$

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