For questions related to valuation functions on a field.

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3
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1answer
49 views

Can a $p$-adic field admit a different valuation?

Let $L/\mathbb{Q}_p$ be a finite extension. Question: Is it possible for $L$ to admit a henselian valuation with residue characteristic $q \not=p$? I would think surely not, but I can't see a ...
2
votes
1answer
308 views

An application of the Weak Approximation theorem - Artin-Whaples Approximation Theorem

Let us recall the weak approximation theorem from Valuation theory in Algebraic Number Theory. Let $K$ be a field, and let $|\cdot|_{1},\dots, |\cdot|_n$ be pairwise non-equivalent nontrivial ...
0
votes
1answer
28 views

What does the power of an ideal *mean*?

I am stumped trying to understand Silverman's definition of $\operatorname{ord}_P(f)$, the (normalized) valuation on $\bar K[C]_P$ (which denotes the localization of a curve $C$'s coordinate ring at ...
0
votes
1answer
17 views

Equivalence of norms in finite dimension over complete fields is true, but false for finite rank modules over complete rings

We know that if $k$ is complete valued field and $V$ a finite dimensional vector space then all norms on $V$ are equivalent. (The field is not necessarily of characteristic $0$ and its absolute value ...
0
votes
1answer
44 views

Extension of a discrete non-archimedean absolute value

Let $P$ be a prime ideal of a Dedekind domain $A$, $v$ an absolute value induced by $P$, and $B$ the integral closure of $A$ in a finite separable extension $E$ of $A$'s quotient field $K$. If $w$ is ...
0
votes
1answer
75 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} ...
10
votes
0answers
234 views

Two discrete valuation rings one of which is contained in another

Let $A$ and $B$ be discrete valuation rings of the same field of fractions. Suppose $A \subset B$. Then $A = B$? I came up with this problem when I was reading van der Waerden's Algebra. The ...
5
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0answers
73 views

Ramification index of infinite primes

I am reading Neukirch's Algebraic Number Theory. On page 184, Chapter 3, Neukirch defines the ramification index of infinite primes as follows: For a finite extension $L/K$ of number fields, and an ...
5
votes
0answers
89 views

Isomorphism of algebras $E_v = \prod_{w\mid v} E_w$

I am currently reading the article Bayer–Fluckiger, Eva, B. Nivedita, and Raman Parimala. "Hasse principle for G–quadratic forms." Documenta Mathematica 18 (2013): 383-392. in which there is a ...
4
votes
0answers
121 views

Determine all discrete valuations on $\mathbb{C}(x)$.

To clarify, for a field $K$, a valuation $v$ on $K$ is a map $v:K^{\times}\to G$ for $G$ an ordered group (written additively) such that for any $a,b\in K^{\times}$: 1) $v(ab)=v(a)+v(b)$; 2) ...
4
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0answers
149 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, ...
4
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0answers
113 views

Transfinite induction and valuation rank

In Engler's valued fields, exercise 3.5.2 goes as follows Construct valuations on $\Bbb{C}$ of rank $\kappa$ for every cardinal $\kappa \leq 2^{\aleph_0}$. The idea behind this (for any ...
4
votes
0answers
67 views

Automorphism of $L|K$ mapping 3 distinct rational points of $S_{L|K}$ to other 3 distinct ones

Let $K$ be a field and consider $L = K(x)$ the field of rational functions. Let $v_{1}, v_{2}, v_{3}$ rational points in the abstract Riemann surface $S_{L|K}$, distinct from each other, and $w_{1}, ...
3
votes
0answers
29 views

An infinite prime can ramify right? (So what is Neukirch talking about?)

I have been under the impression for several years that if $L/K$ is an extension of number fields, then an infinite place of $K$ is said to ramify in $L$ if it comes from a real embedding of $K$ which ...
3
votes
0answers
22 views

Is $\mathcal O^{\ast}/U^{(n)}\cong (\mathcal O/\mathfrak p)^{\ast}$ as topological groups?

I have the following: $K$ is a field with discrete valuation $v$, $\mathcal O$ its valuation ring and $\mathfrak p$ the maximal ideal and $U^{(n)}=1+\mathfrak p^n$ the $n$-th unit group for $n\geq 1$. ...
3
votes
0answers
41 views

Ramification group of valuations - need terminology

I am lost and need some terminology (also hopefully sources). Let $L/K$ be a Galois extension, and $w$ be a valuation of a $L$, lying above a valuation $v$ of $K$. Notice that I do not suppose that ...
3
votes
0answers
67 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 ...
3
votes
0answers
103 views

Question about complete DVR.

Is there a simple proof that you know to the following statement: If the residue field $k$ of a complete DVR $R$ has the same characteristic as $R$, then $R$ contains a subfield isomorphic to $k$. ...
3
votes
0answers
91 views

Question about localizations of discrete valuation rings.

Let $R$ be a discrete valuation ring. Then $R$ has only two prime ideals: $0$ and the maximal ideal $\mathfrak{m}$. It is said in Hartshorne, page 74, Example 2.3.2 that the localization of $R$ at $0$ ...
2
votes
0answers
22 views

How to prove that any infinite algebraic extension of a complete field is never complete?

My first idea is using Baire category theorem since I thought an infinite algebraic extension should be of countable degree. However, this is wrong, according to this post. This approach may still ...
2
votes
0answers
41 views

The number of abolute value on $\mathbb{Q}(\sqrt{2})$

Let $|.|$ be the usual absolute value on $\mathbb{Q}$. The number of absolute value on $Q(\sqrt{2})$ extending |.| is 2 since $x^2-2=(x-\sqrt{2})(x+\sqrt{2})$ in $\mathbb{R}[x]$. Let ...
2
votes
0answers
67 views

Equivalent Hensel's lemma?

Let $K$ be a complete field with nonarchimedian valuation $|\cdot|$ and $\mathcal{O}_K := \{x \in K: \; |x| \le 1\}$, $\mathfrak{m} := \{x \in K: \; |x| < 1\}$. I have seen two statements that do ...
2
votes
0answers
75 views

How to prove that a DVR is not complete

My question is inspired by a comment in this topic. How to prove that $R=\mathbb C[x]_{(x)}$ is not complete in the topology of its maximal ideal? One knows that $R$ is a DVR, and its field of ...
2
votes
0answers
52 views

A question on a sum of valuations

Let $A$ be a discrete valuation ring of characteristic zero. Let $v$ be the valuation on $A$. Let $I$ be a finite index set and $d_i$ a positive integer for all $i$ in $I$ and define $$ d:= \sum_{i ...
1
vote
0answers
27 views

Geometric structure on the set of valuation rings of a field

Let $K$ be a field. Let $\mathcal{O}_K$ be the intersection of all valuation rings with quotient field $K$. Can someone give an example of a field $K$ in which we don't have a bijection of sets: $$ ...
1
vote
0answers
10 views

Why is the separable closure of a field in it's completion not complete.

I am currently reading "Algebraic Number Theory" By Neukirch and I am a bit confused by what is here. It is on page 143, it states this, $(K,v)$ is a nonarchimedian valued field and ...
1
vote
0answers
26 views

Surjectivity of the derivation map on Washnitzer Algebra

Let $K$ be a non-archimedean field of characteristic zero and $||.||:K\to \mathbb{R}_{\geq 0}$ be its absolute value. Define the Washnitzer Algebra as: $$W_n=\{\sum_{u\in \mathbb{Z}_{\geq 0}^n} \in ...
1
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0answers
45 views

Extension of valuation

This is a question from Milne's 'Algebraic Number Theory'. Let $K$ be a valued field with absolute value $|\cdot|$ and $L=K(\alpha)$ a finite separable extension of $K$. Let $\hat{K}$ be the ...
1
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0answers
29 views

Computing other valuations of a field

Assume $k$ is an algebraically closed field, and $x$ and $y$ are indeterminant over $k$, I know valuations of $F$, the field of fractions of the ring $A=\dfrac{k[x,y]}{I}$ where $I$ is an ideal of ...
1
vote
0answers
32 views

Trivial absolute value

Let $K/L$ be a algebraic extension. Suppose that $\left|\cdot\right|$ is a absolute value in $K$ such that is trivially in $L$. Then is trivially in $K$. Thanks for anny suggestion. If is trivially ...
1
vote
0answers
50 views

An example of a henselian valuation of rank 2.

I need to know simple examples of valuations of rank bigger than one. Please help me to concrete some examples of valuations of rank bigger than one with their valuation rings (specially henselian ...
1
vote
0answers
23 views

A valued field is complete iff its ring of intergers is complete

Let $K$ be a field, and let $v$ be a valuation defined on K, and let $O$ be the ring of integers, and $M$ be the maximal ideal. How to show that $K$ is complete (i.e $K$ is equal to its completion) ...
1
vote
0answers
121 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 ...
1
vote
0answers
49 views

A trace formula for an extension of complete valued fields

I'm trying to understand the following fact. Proposition. Let $L/K$ be a finite separable extension of complete valued fields, with respective valuations $w$ and $v$, valuation rings $\mathcal{O}_L$ ...
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0answers
76 views

Henselization and valued field

What is the importance of henselization in valuation theory, when the rank of valuation is bigger than one? Thanks
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0answers
23 views

Definable valuation ring

If $(K,v)$ is henselian and $O_v$ is $\phi$-$\text{definable}$, why do I have that if $L\equiv K$ (in the language of ring) then $L$ admits a non-trivial henselian valuation ring? I understand if ...
0
votes
0answers
21 views

unramified extension of valued fields

I came across the following exercise: Let $M$ be a valued field with subfields $E$ and $L$, and suppose that $L$ is finite over some field $K\subseteq L\cap E$. Show that $EL/E$ is unramified if ...
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0answers
58 views

Roots of unit in a DVR

Let $A$ be a DVR such that its fraction field $K$ is complete w.r.t to the natural absolute value in $K$. I am trying to prove that the projection from $A$ to the residue class field, $F$, maps the ...
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votes
0answers
18 views

Reference request for valuations in algebraic number theory

I am going through the book "Primes of the form $x^2$ + n$y^2$" and I understand the background material about algebraic number fields and class numbers(from Ireland and Rosen). However, I do not have ...
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votes
0answers
13 views

Determine the valuation of $\rho$ with $F = k(x,\rho)$ at the only place above $(\infty)$.

Let us assume that we have the following setup. Let $F=k(x,\rho)$ be an algebraic function field with $$f(x,y) = y^n+a_1y^{n-1}+\cdots+ a_iy^{n-i}+\cdots+a_n \in k[x][y]$$ irreducible in $y$, and ...
0
votes
0answers
19 views

How to construct a unique valuation for $k\left(T_{i}\right)_{i\in\mathbb{N}}$ in $\mathbb{Z}^{\left(\mathbb{N}\right)}$?

Let $k$ be a field and $\left(T_{n}\right)_{n\in\mathbb{N}}$ indeterminates over $k$. Let $K=k\left(T_{n}\right)_{n\in\mathbb{N}}$ and $\varGamma:=\mathbb{Z}^{\left(\mathbb{N}\right)}$ the abelian ...
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0answers
39 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 $\mathcal{O}$ ...
0
votes
0answers
41 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 ...
0
votes
0answers
33 views

Criterion of separability of function on a curve.

Let $K$ be finitely generated extension of an algebraically closed field $k$ of transcendence degree $1$, $\operatorname{char }k =p$. Let $C(K)$ be set of discrete valuation rings $(\mathcal ...
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votes
0answers
66 views

Application of Hensel's lemma

Show that the polynomial $\Phi(x)=x^2 -2 \in O(\widehat{\Bbb Q_2})[x] $ has no root in $\widehat{\Bbb Q_2}$, even though $\bar\Phi(x)\in E(\widehat{\Bbb Q_2})[x]$ has a root in $E(\widehat{\Bbb Q_2}) ...