4
votes
1answer
35 views

Lattice in a vector space of dim 2 over a valuated field.

I'm reading "Arbres, amalgames et SL2" of J.P. Serre, and something is not clear to me, but is to him :) Let $k$ be a field, with a discrete valuation $v$, ie a group epimorphism $v:k^\ast \to ...
1
vote
1answer
44 views

Embedding $\mathbb Q^c$ into $\mathbb Q^c_p$

Let $p$ be a prime number and $\mathbb Q_p$ the $p$-adic completion of $\mathbb Q$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. Is there an embedding $$j: \mathbb Q^c ...
7
votes
1answer
115 views

A valuation-like function $w: \mathbb{N}^{+} \rightarrow \mathbb{N}$ is a $p$-adic valuation?

This question is a variant of problem 4, pg. 21, from Birkhoff and Maclane, A Survey of Modern Algebra. Given a function $w: \mathbb{N}^+ \rightarrow \mathbb{N}$ that behaves like a valuation ...
1
vote
1answer
27 views

A certain ideal of a valuation ring

A question from Frohlich and Taylor's book 'Algebraic Number Theory', page 60. Let $\mathfrak o$ be a Dedekind domain with quotient field $K$ and let $v$ be a discrete valuation on $K$. Let ...
0
votes
1answer
28 views

Two discrete valuations on a Dedekind domain

A question from Frohlich and Taylor's book 'Algebraic Number Theory', page 60. Let $\mathfrak o$ be a Dedekind domain with quotient field $K$ and let $v$ be a discrete valuation on $K$. Let ...
0
votes
2answers
73 views

Primes corresponding to embeddings of a number field

Let $k$ be a number field. Define a prime of $k$ to be an equivalence class of absolute values on $k$. If $\sigma:k\hookrightarrow \mathbb{C}$ is an embedding of $k$ into the complex numbers then we ...
2
votes
1answer
42 views

Integral elements of a non-archimedean completion of a number field

Let $k$ be a number field, $v$ a discrete non-archimedean valuation on $k$. Let $k_v$ be the completion of $k$ wrt $v$ and $\mathcal O_v$ its valuation ring. My question is: If $x\in k_v$ then is ...
1
vote
1answer
26 views

Quotients of a valuation ring in the completion of a number field

Let $k$ be a number field, $v$ a discrete (non-archimedean) valuation on $k$. Let $k_v$ be the completion of $k$ with respect to $v$. Also let $\mathcal{O}_v$ be the valuation ring of $k_v$ and ...
0
votes
0answers
59 views

Extensions of valuations

I'm trying to understand how a valuation $v$ of a field $K$ extends to an algebraic extension $L$. In chapter 8 of his book ANT, Neukirch first chooses a $K$-embedding $\tau : L \longrightarrow ...
2
votes
2answers
60 views

Valuations, Isomorphism, Local ring

Let $x \in \mathbb{Q}, \, x \neq 0$, such that $x=p^r \frac{a}{b}, \quad a,b, r \in \mathbb{Z}, \quad p \nmid a, \quad p\nmid b$. Let $v_p(x):=r$ and $v_p(0):= \infty$. Also, $$\mathcal O_p= \left\{ ...
1
vote
0answers
145 views

units in discrete valuation rings

Let $B/A$ be an extension of discrete valuation rings such that the purely inseparable extension of residue fields $l/k$ has a primitive element, i.e., $l=k(y)$ for some element $y$ in $l$. I want to ...
0
votes
1answer
61 views

How do I compute the discrete valuation of the sum of two elements

Let $O$ be a discrete valuation ring with valuation $v$. We normalize by $v(\pi) =1$, with $\pi$ a prime element in $O$. By definition, for all $x,y$ in $O$, we have $v(x+y) \geq \min (v(x),v(y))$. ...
1
vote
2answers
233 views

A characterisation of tame ramification

The following is the statement from Algebraic Number Theory by Neukirch (Chapter 2 Proposition(7.7) p155) Blockquote Suppose $K$ is Henselian field, $p=char(\kappa)$ , the character of the ...
3
votes
1answer
87 views

Dividing the ramification index of an extension

Let $f(x)$ be a polynomial of degree $m$ over $\mathbb{Q}_{p}$ with all roots $r_{i}$ such that $\operatorname{ord}_{p} r_{i} = \frac{1}{p}$. Why does $p$ have to divide the ramification index of ...