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Questions tagged [valuation-theory]

For questions related to valuation functions on a field.

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4answers
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Examples of Non-Noetherian Valuation Rings

For valuation rings I know examples which are Noetherian. I know there are good standard non Noetherian Valuation Rings. Can anybody please give some examples of rings of this kind? I am very ...
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1answer
138 views

Simple property of a valuation on a field

Let $F$ be a field and $v:F\rightarrow G\cup\{\infty\}$ be a valuation on $F$ so $G$ is a totally ordered abelian group with $\infty$ having the properties $\infty+\infty=g+\infty=\infty+g=\infty$ and ...
2
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2answers
718 views

Does every non-Archimedean absolute value satisfy the ultrametric inequality?

The Archimedean property occurs in various areas of mathematics; for instance it is defined for ordered groups, ordered fields, partially ordered vector spaces and normed fields. In each of these ...
6
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1answer
301 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, ...
3
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1answer
225 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 $v(x+y) \geq \min (v(x),v(y))$. ...
3
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1answer
114 views

Absolute values equivalence in $\mathbb{F}_{q}(x)$

Let $q=p^n$ a prime power, $f(x)$ a monic irreducible polynomial in $\mathbb{F_q}[x]$ and $q_f = \text{the size of the residual field of f.}$ Define the following absolute values over $\mathbb{F}_q(x)$...
2
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1answer
168 views

Can we have a perfect field of characteristic $p$ with a non-archimedean valuation?

Motivation for the question: in Neukirch - Algebraic Number Theory (Chapter II - Section 6 - Henselian Fields) the author discusses the context of a nonarchimedean valued field $K$ and its completion $...
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2answers
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Concrete examples of valuation rings of rank two.

Let $A$ be a valuation ring of rank two. Then $A$ gives an example of a commutative ring such that $\mathrm{Spec}(A)$ is a noetherian topological space, but $A$ is non-noetherian. (Indeed, otherwise $...
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2answers
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Why is the restricted direct product topology on the idele group stronger than the topology induced by the adele group?

I am confused by the saying that the restricted direct product topology on the idele group is stronger than the topology induced by the adele group. And perhaps this is elaborated by the following ...
4
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2answers
218 views

ring of adeles: ring of integers or valuation ring

I have read different definitions of the ring of adeles: The ring of adeles is defined as the restricted topological product of the completions $K_v$ of a number field $K$ either with respect to ...
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0answers
299 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 ...
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0answers
152 views

A slightly stranger Hensel's Lemma

I'm trying to understand the solution to this problem. It came up when doing some revision. It is essentially to show that the conclusion of Hensel's Lemma holds if we have take a valuation ring $R$ ...
3
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1answer
1k views

How do we find the prime ideals of a ring of integers of a number field?

How do we find the prime ideals of a ring of integers of a number field? For example, for $F=\mathbb Q(\sqrt{-5})$ the ring of integers of $F$ is $\mathbb Z[\sqrt{-5}]$ (since $-5\equiv3 \pmod 4$). ...
8
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1answer
419 views

Two discrete valuation rings, one contained into 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 ...
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0answers
143 views

Is the topology of the p-adic valuation to the unramfied extension discrete?

Consider $\mathbb Q_p^{\text{ur}}$ the maximal unramified extension of the p-adic numbers. Suppose that on $\mathbb Q_p$ we have the usual absolute value that extends $|\frac{a}{b}|_p=\frac{1}{p^{v_p(...
5
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1answer
65 views

Is the result true when the valuation is trivial and $\dim(X)=n$?

Here I proved the following result: Proposition: Let $(K,|\cdot|)$ be a valued field with non-trivial valuation and let $X$ be a vector space over $(K,|\cdot|)$. Two norms $p_1,p_2$ on $X$ are ...
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0answers
586 views

Archimedean places of a number field

Let $K$ be a number field with an Archimedean absolute value $|\cdot |$ and let $\bar{K}$ be the completion of $K$ wrt this valuation. Then $\bar{K}\cong \mathbb R $ or $\mathbb C$. My question is: ...
2
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1answer
531 views

Formal power series ring over a valuation ring of dimension $\geq 2$ is not integrally closed.

I recently tried exercise 10.4 in Matsumura's Commutative Ring Theory, but got stuck. The question is: If $R$ is a valuation ring of Krull dimension $\geq 2$, then the formal power series ring $R[[...
7
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1answer
173 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 ...
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2answers
829 views

Is every local ring a valuation ring?

Is every local ring a valuation ring? I know the answer is no and the first example comes to my mind was as following (I started with smallest fields, as $\mathbb{Z}_2$ and $\mathbb{Z}_3$ are not ...
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1answer
495 views

How to show the only absolute value on a finite field is the trivial one.

Define the trivial absolute value $|\cdot|$ by $|x| = 1$ if $x \neq 0$ or $|x| = 0$ if $x=0$. The textbook I'm currently reading (Gouvêa - P-adic Numbers An Introduction) asked me to show that for a ...
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3answers
386 views

The ring of integers of a number field is contained in any valuation ring.

Let $K$ be a number field and let $v$ be a surjective discrete valuation on $K$. Let $\mathcal O_v$ the valuation ring of $v$, namely $$\mathcal O_v=\{x\in K\,: v(x)\ge 0\}$$ Let's denote with $O_K$ ...
3
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2answers
862 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 ...
3
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1answer
111 views

How do we extend the valuation on $K[x]$ to a valuation on $K(x)$?

Given that $v$ is a non-archimedean valuation on $K$, we can extend it to $|\cdot|:K[x]\to\mathbb{R}$ by $|a_0 + a_1x + \cdots +a_nx^n|=\max\left\{|a_1|,\ldots,|a_n|\right\}$. My question is how can ...
2
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1answer
209 views

Existence of an element of given orders at finitely many prime ideals of a Dedekind domain

Let $A$ be a Dedekind domain. Let $P$ be a non-zero prime ideal of $A$. Let $\alpha \in A$. Let $k$ be a non-negative integer. If $\alpha \in P^k$ and $\alpha\notin P^{k+1}$, we write $v_P(\alpha) = k$...
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2answers
96 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\{ ...
2
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1answer
56 views

Proof of valuation property (function on $K(X_1,\,\dots,\, X_n)$)

Let $K$ be a field and consider the field of rational multivariate polynomials $K(X_1,\,\dots,\, X_n)$ in $K$ for some $n \geq 1$. Define $\mathbb{P}:= (0,\, \infty)$ as the set of positive reals and ...
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2answers
426 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 ...
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1answer
424 views

Localization of a valuation ring at a prime is abstractly isomorphic to the original ring

Let $A$ be an integral domain with quotient field $K$. We say that $A$ is a valuation ring if for any $0 \neq x \in K$, $x$ or $\frac{1}{x}$ lies in $A$. Then $A$ is necessarily a local ring. If $A$...
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2answers
315 views

How many absolute values are there?

My question is the following: Are there algebraic norms on the fields $\mathbb{R}, \mathbf{Q_p}$ ''other'' than the absolute value, respectively $|\cdot|_p$? Now phrasing more precisely: If generally ...
3
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1answer
76 views

Valuation rings and total order

Let $K$ be a field and $\mathcal{O}$ be a valuation ring of $K$ (So $\mathcal{O}\subset K$ is an integral domain with the property that either $x$ or $x^{-1}$ is in $\mathcal{O}$ for all $x\in K$). ...
2
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1answer
77 views

If $v$ is a valuation and $v(x)<v(y)$ then $v(x+y)=v(y)$. [duplicate]

I was studying from some book and I came across something I haven't been able to justify. Let's suppose se have a field $K$ and an ordered group $G$ (with multiplicative notation) with an extra ...
2
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1answer
78 views

How to reconstruct a valuation from a valuation ring

I have been independently studying valuation rings for a research paper for an undergraduate abstract algebra course. I've been heavily relying on Gathmann's lecture notes, found at http://www....
2
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1answer
200 views

There are no archimedean function fields

Definition: a field $L\supseteq K$ is called a function field over $K$ if the extension $L|K$ is finitely generated, regular and of transcendence degree $1$. In the book "Topics in the theory of ...
2
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1answer
133 views

Closure of an ideal with respect to p-adic valuation

Goro Shimura, Euler Products and Eisenstein Series, Chapter II (Adelization of algebraic groups and automorphic forms) "Let $\mathbf{F}$ be an algebraic number field of finite degree. We denoe by $\...
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2answers
113 views

Is the topology of the p-adic valuation to the unramfied extension complete?

Consider $\mathbb Q_p^{\text{ur}}$ the maximal unramified extension of the p-adic numbers. Suppose that on $\mathbb Q_p$ we have the usual absolute value that extends $|\frac{a}{b}|_p=\frac{1}{p^{v_p(...
1
vote
2answers
180 views

Discrete valuation ring and finitely generated submodules

Let $R$ be a Discrete Valuation Ring with fraction field $K$. Will this imply any proper $R$-submodule of $K$ is finitely generated (hence a fractional ideal)? I know $K$ is not finitely generated as ...
0
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1answer
111 views

Can a valuation ring properly contains another valuation ring with the same field of fractions?

Definition of valuation ring: Let $R$ be an integral domain with $frac(R)=K$. Then $R$ is said to be a valuation ring if (1) $R \neq K$ (2) $\forall x \in K, x \in R$ or $x^{-1} \in R$. Now my ...
0
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1answer
78 views

Discrete Valuation Rings with property that $v(x+y)=\min(v(x),v(y))$ [duplicate]

Let $R$ be a discrete valuation ring on a fraction field $K$ of $R$. If $x,y\in K$ such that $v(x)< v(y)$, prove that $v(x+y)=\min(v(x),v(y))$. By definition of discrete valuation, we have that ...
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1answer
65 views

When do we have equality in the third axiom of valuations? [duplicate]

Let $R$ be a ring and $v: R \to \mathbb Z \cup \{+\infty\}$ a map that meets following axioms: $v(a) = +\infty \iff a=0$ $v(ab) = v(a)+v(b)$ $v(a+b) \geq \min\{v(a),v(b)\}$ I have to show that $v(a) ...