# Questions tagged [valuation-theory]

For questions related to valuation functions on a field.

40 questions
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 ...
1answer
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### 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 ...
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 ...
1answer
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### 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, ...
1answer
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### 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 $v(x+y) \geq \min (v(x),v(y))$. ...
1answer
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### 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 $\mathbb{F}_q(x)$...
1answer
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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 ...
2answers
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### 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 ...
0answers
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### 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 ...
0answers
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### 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 $R$ ...
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 $-5\equiv3 \pmod 4$). ...
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 ...
0answers
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1answer
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### 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 ...
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 ...
1answer
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### 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 ...
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 $O_K$ ...
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 ...
1answer
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### 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 ...
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 $v_P(\alpha) = k$...
2answers
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### 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\{ ...
1answer
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### 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 ...
2answers
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### 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 ...
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 $A$...
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 ...
1answer
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### 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 $x\in K$). ...
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 ...
1answer
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### 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....
1answer
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### 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 ...
1answer
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2answers
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### 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 ...
1answer
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### 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 ...
1answer
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### 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 ...
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) ...