All Questions
7 questions with no upvoted or accepted answers
8
votes
0
answers
245
views
Relation between valuation of p-adic regulator of totally real field and its finite p-unramified abelian extensions
For each prime number $p$ and number field $k$, there exists at least one extension $k_{\infty}/k$ with Galois group isomorphic to $\mathbb{Z}_p$, the cyclotomic $\mathbb{Z}_p$-extension. If $k_p/k$ ...
- 1,521
7
votes
0
answers
487
views
The character of a separable degree-$p$ extension of a local field of residual characteristic $p$ ?
Let $p$ be a prime number and $F$ a finite extension of ${\mathbf Q}_p$ or of ${\mathbf F}_p((t))$. I'm going to define a natural map from the set ${\mathcal S}_p(F)$ of degree-$p$ separable ...
- 18.7k
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
4
votes
0
answers
144
views
Does a countably generated $\mathbb{Q}$-algebra inject into some $p$-adic field?
Let $K$ be a subfield of $\mathbb{C}$. If $K$ is finitely generated over $\mathbb{Q}$, then $K$ injects into $\mathbb{Q}_p$ for some $p$.
Assume that $K$ is countably generated, i.e., $K= \...
- 71
2
votes
0
answers
92
views
Is the extension field by zeros of $x^{2m}-p^{2(m-1)}=0$ over $\mathbb Q_p$ totally ramified?
Let $m \geq 2$ be an integer. Consider the polynomial $f(x)=x^{2m}-p^{2(m-1)} \in \mathbb{Q}_p[x]$.
I want to study the field extension by the zeros of $f(x)$ over $\mathbb{Q}_p$.
What is the degree ...
- 930
2
votes
0
answers
257
views
Is the absolute Galois group $\text{Gal}(\bar K/K)$ isomorphic to $\text{Gal}(K(S)/K)$?
Let $K$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}$, maximal ideal $\mathfrak{m}$ and uniformizer $\pi$. Let $\bar K$ be the algebraic closure of $K$ and $\bar{\...
- 930
0
votes
0
answers
202
views
When is $u \circ v=v \circ u$ for $p$-adic power series $u$ and $v$ in two power series rings $A$ and $B$ respectively?
Let $K \supset \mathbb{Q}_p$ be the $p$-adic field with ring of integers $O_K$ and maximal ideal $m_K$. Let $\bar K$ be the algebraic closure and $\bar{m}_K$ be the integral closure of $m_K$ with ...
- 930