0
votes
0answers
22 views

Question about valuation rings of a rational function field

Let $k$ be a field and $f \in k[x,y] \setminus k$. Write $E=k(f)$ and $L=k(x,y)$. Assume that there exists a $g \in k(x,y)$ such that $L = E(g)$. Suppose there exists a valution ring $\mathcal{O}$ ...
3
votes
0answers
45 views

Find valuation rings of the function field $k(x,y)/k(xy)$ which do not contain $k[x,y]$.

Let k be any field, then how does one find valuation rings of the function field $k(x,y)/k(xy)$ which do not contain $k[x,y]$? I believe there are two. If we consider a rational function field ...
0
votes
2answers
35 views

Discrete valuations of the rational numbers

I'm trying to find every discrete valuation on the field of rational numbers. If $a\in \mathbb Q$, we can write $a=p^j\frac{x}{y}$, where $p$ is a prime number and $p\nmid x$ and $p\nmid y$. We can ...
2
votes
1answer
41 views

Places of this extension

I'm reading this book. I'm trying to find the degree of the places of the extension $\mathbb C(X)\mid\mathbb R$. I know the places of the extension $\mathbb R(X)\mid\mathbb R$ and I've already ...
3
votes
1answer
60 views

real closure of an archimedean field

my question is: Is an archimedean field dense in its real closure? I know that in the non-archimedean case, this does not have to be true (e.g., rational fucntions). Thanks!
0
votes
0answers
24 views

Hensels Lemma in many variables

Let $(K,v)$ be a henselian valued field, with valuation ring $\mathcal{O}$ and residue field $Kv$. Then given a polynomial $f \in \mathcal{O}[x]$, henselianity tells that given some suitable ...
4
votes
1answer
42 views

Why is the order of a prime element well-defined?

This is in relation to the $p$-adic valuation on the field of fractions $F$ of an integral domain $D$. The idea is that for each $x \in F$ there is a unique maximal $k$ such that $x = p^k u v^{-1}$ ...
0
votes
1answer
35 views

Is the residue field of an algebraically closed field with respect to a non-trivial valuation infinite?

Let $K$ be an algebraically closed field with a non-trivial valuation. Let $\Bbb k$ be the residue field, i.e. $\Bbb k = R/\mathfrak m$ where $R$ is the valuation ring $R:=\{a\in K\mid \text{val} ...
4
votes
1answer
53 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 ...
0
votes
0answers
16 views

Division in a complete subring of a local field

Suppose $A$ is a complete subring of a local field such that a prime element $\pi$ belongs to $A$. Is it true that if $\beta=\pi^k u$ (with $k\ge 0$ and $v(u)=0$) and $\beta\in A$ then also $u\in A$? ...
5
votes
1answer
108 views

Valuation ring in an algebraically closed field

Let $k$ be an algebraically closed field and $(R, \mathfrak{m})$ a valuation ring in $k$ i.e. the field of fractions of $R$ is $k$. Then Mumford (Red Book, page 127) claimed that the residue field ...
2
votes
2answers
142 views

Local fields and infinite extensions, basic questions

Notation throughout: Let $K$ be a discrete valuation field and $L/K$ an infinite (not necessarily Galois) extension of $K$. 1) How can/does one define a ramification index $e(L/K)$ for $L/K$? It ...
4
votes
0answers
128 views

Intuition behind “Non-Archimedean” — two senses of “non-archimedean”.

There appear to be two senses of the qualifier "Archimedean" for fields. One is for ordered fields, and one is for "valued fields" (fields with an absolute value function defined). In the first case, ...
2
votes
2answers
163 views

The composite of all unramified extensions inside an algebraic closure

I'm reading Ch.II, $\S$ 7 of Neukirch's Algebraic Number Theory and I'd be really grateful if someone could help me understand the following: Let $K$ be a complete valued field wrt a non-archimedean ...
3
votes
1answer
279 views

Non-Archimedean Fields

Are there non-Archimedean fields without associated valuation or being a non-archimedan field implies it is a valuation field? I understand that a non-Archimedean field is a field which does not ...
0
votes
1answer
148 views

What is a valuation associated to an ordering on a field?

If $(K,\leq)$ is a totally ordered field with $P\!=\!\{\alpha\!\in\!K;\, 0\!\leq\!\alpha\}$, how is the valuation associated to $P$ defined? I was searching through Prestel & Delzell's Positive ...
0
votes
0answers
53 views

What does it mean for a valuation to be normed?

I have a homework problem that uses the term: "a discrete normed non-trivial valuation" on a field. We've defined the discrete trivial valuation in class, so that part is clear. I think the natural ...
5
votes
3answers
596 views

Algebraic Closure of Puiseux Series

Using the construction $R_N = K[t^\frac1N]$ $L_N = Quot(R_N)$ and $P = \bigcup_{N\in \mathbb{N}} L_N$ one automatically gets that the puiseux series are a field. Nevertheless they are also an ...
13
votes
1answer
337 views

Why is $\mathbb{C}_p$ isomorphic to $\mathbb{C}$?

I know that two closed fields of caracteristic $0$ and uncountable are isomorphic iff they have the same cardinality. But I don't know why $\mathbb{C}_p$ has the same cardinality as $\mathbb{C}$. Can ...