All Questions
7 questions
4
votes
1
answer
367
views
Conductor and local Kronecker–Weber theorem
Given an abelian extension $K$ of $\mathbb{Q}$, the global Kronecker–Weber theorem tells us that there exist a positive integer $N$ and a primitive $N$-th root of unity $\zeta_N$ such that $K\subseteq ...
- 549
4
votes
0
answers
170
views
Does $p$-adic Baker theorem holds in the given case?
Let $p$ be a prime number, $\mathbb{Q}_p$ the field of $p$-adic numbers, and $\mathbb{C}_p$ the completion of the algebraic closure of $\mathbb{Q}_p$. Let $U_p$ be the units $(1+\mathfrak{m})$ of $\...
- 930
3
votes
0
answers
164
views
Using the Hilbert symbol to find nice field extensions
Let $p$ and $q$ be (not necessarily distinct) odd primes and let $F=\mathbb{Q}_p(\mu_q)$. The $q^{th}$ Hilbert symbol induces a non-degenerate alternating form $$(\cdot,\cdot)_q:F^\times/(F^\times)^q\...
- 2,516
1
vote
1
answer
190
views
Hilbert symbols vanishing
Let $p$ be an odd prime, and let $E/\mathbb{Q}_p$ be a finite extension that contains a primitive $p$-th root of unity $\zeta_p$ but not a primitve $p^2$-th root of unity $\zeta_{p^2}$. Let $a,b \in E^...
- 11.3k
2
votes
0
answers
166
views
Relative Leopoldt defect
Let F be a totally real number field such that the Leopoldt conjecture holds at a prime number $p$ and $M$ be a quadratic totally real extension of $F$.
Is there a bound of the Leopoldt defect of $M$ ...
- 1,066
1
vote
0
answers
146
views
Class field theory, Ideles class
Let $H$ be a totally complex Galois extension of $\mathbb{Q}$ and $g:G_H \rightarrow \bar{\mathbb{Q}}_p$ be a continuous morphism. By class field thoery we have $\mathrm{Hom}(G_H, \bar{\mathbb{Q}}_p)\...
- 1,066
5
votes
0
answers
758
views
maximal abelian extension of quadratic extension of $\mathbb Q_p$
I read this article "Local class field theory via Lubin-Tate theory" http://arxiv.org/pdf/math/0606108v2.pdf. And I want to find the maximal abelian extensions for quadratic extensions of $\mathbb Q_p,...
- 51