Questions tagged [valuation-theory]

For questions related to valuation functions on a field, and their corresponding valuation rings.

A valuation is a function on a field that provides a notion of size or multiplicity for elements of a field. More specifically, a valuation is a surjective function from the unit group of a field to an ordered abelian group. An example of a valuation is the $p$-adic valuation on $\Bbb{Q}$. A field with a valuation on it is known as a valued field. Valuations are very useful tools that are often used in the study of algebraic geometry and algebraic number theory. Some topics in valuation theory include valuation extensions, such as Chevalley's extension theorem, and Henselian fields.

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Classical number theoretic applications of the $p$-adic numbers

I am sure we can all agree that the $p$-adic numbers are highly fascinating objects in their own right - just as the closely related theory of valuations. Having independently read up on the $p$-adic numbers for a few weeks now, I have so far only…
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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 anyone give me some reference or hint ?
user10676
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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 $A$ would be a discrete valuation ring.) Is there…
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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 eager to know. Thanks.
GA316
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Why does the equation $x^2-82y^2=\pm2$ have solutions in every $\mathbb{Z}_p$ but not in $\mathbb{Z}$?

I have been working on an exercise in H. P. F. Swinnerton-Dyer's book, A Brief Guide to Algebraic Number Theory. The question is like this: Show that $x^2-82y^2=\pm2$ has solutions in every $\mathbb{Z}_p$ but not in $\mathbb{Z}$.What conclusion…
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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 extends to complex embeddings of $L$. I am…
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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 infinite prime $\mathfrak{B}$ lying over…
Zeyu
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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 finite field $\mathbb{K}$ the only possible…
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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 motivation is a follows. Let $A$ be a Noetherian…
Makoto Kato
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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 interesting so I came to next possible one, means…
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Definition of algebraic variety

In general, the definition of an algebraic variety differs from one reference to other. The definition that I was used to is to consider an algebraic variety as an integral scheme of finite type. However this definition differs from the one in the…
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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 $F$ is a field, an algebraic norm is a map…
Fabian Werner
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Applications of valuation rings

Some background: I am in the process of writing a research paper for an undergraduate abstract algebra course. I've chosen to write my paper on valuation rings and discrete valuation rings. The goal of the paper is to broaden my own and my…
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equivalent characterizations of discrete valuation rings

Let $R$ be a commutative ring with identiy, then the following are equivalent: $R$ is a DVR $R$ is a local Euclidean domain that is not a field. $R$ is a local PID that is not a field. $R$ is a local Dedekind domain that is not a field. $R$ is a…
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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 contexts it is roughly the following…
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