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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.

721 questions
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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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Extension of valuation

We define a valuation on the field of rational number $\mathbb Q$ as follows. For example if we choose a prime number $2$ then for $x \neq 0\in \mathbb Q$, $v(x) = v(2^{n}a/b)= n$ where $n$ is an integer and $a$ and $b$ are relatively prime integer.…
Rajesh
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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 we extend $|\cdot|$ to a valuations on $K(x)$?…
Galois
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$\mathrm{Spa}(A,A)=\{A\to V \text{ valuation ring}\}$ modulo faithfully flat maps

Let $A$ be a discrete ring, and $\mathrm {Spa}(A,A)$ the space of (continuous) valuations on $A$, up to equivalence, which take values $\le1$ on all of $A$. In Clausen–Scholze’s Condensed Mathematics notes (page 64, just after Definition 9.5) it is…
Hermetically Sealed Halibut
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Definition of absolute value

Usually absolute value $v$ on a field $K$ is defined as a map $x\to |x|_v$ such that $\forall x,y\in K$ one has $$\begin{align} &|x|_v=0\iff x=0\\ &|xy|_v=|x|_v|y|_v\\ &|x+y|_v\leq |x|_v+|y|_v \end{align}$$ If one has $|x+y|_v\leq \max(|x|_v,|y|_v)$…
marwalix
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Valuation on the field of rational functions

I'm currently learning the basic machinery of valuation theory from "ordered exponential fields" by Kuhlmann, and I have a question regarding an assertion made on page 1. She gives the standard axioms for a valuation…
Alec Rhea
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Algebraically maximal valued fields

Does anyone know of an elementary proof that an algebraically maximal field is Henselian (ie one that does not assume knowledge of henselizations)? Definitions: We say a valued field $(K,v)$ is algebraically maximal if it has no proper algebraic…
Conrad
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Non-archimedean valuation

In the definition of non-archimedean valuation the value group is the group of real numbers $\mathbb R$. Can we replace $\mathbb R$ by any totally ordered group different from $\mathbb R$ ? Thanks
Sharma Sharma
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we have min-valuation?

http://mathworld.wolfram.com/Valuation.html absolute value satisfy triangle inequality $p$-adic valuation satisfy ultrametric inequality $|x+y| ≤ \max\{|x|,|y|\}$ then can we find some valuation satisfy $0≤|x|$ $|x|=0$ iff $x=0$ $|xy|=|x||y|$ $|x+y|…
flower
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why $-\log(x) $ is a valuation like $v_p(x)$?

Why $\log$ is used in the definition of valuation? I can not understand how $\log(x)$ gives rational value always. We know that p-adic valuation of $x \in \Bbb Q$ is given by $$|x|_p=p^{-v_p(x)}.$$ We also define valuation to be $-\log(x)$. Also we…
MAS
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Isomoprhism in complete field $(k,v)$

Let $(k,v)$ be any complete field and $U = \{x | x \in k^{\times}, v(x)=0 \}$. Let $\langle \pi \rangle $ denote the cyclic subgroup of $k^{\times}$, generated by a prime element $\pi$. My question is why isomorpism $k^{\times}/U \simeq…
Hikicianka
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Suppose that the spot price...

Suppose the spot price of gold is \$300 per ounce and the risk-free interest rate for one year is 5%. What is a reasonable value for the one-year forward price of gold? The answer is \$315, right? Suppose the one-year forward price of gold is…
Nicolas Cloet
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Units in discrete valuation ring

Let $k$ be field and $K=k(x)$ the field of rational function functions over $k$. Suppose that $p(x)\in k[x]$ is an irreducible polynomial. Define a map $v:K\to\mathbb Z$ by $v(p^r\frac{m}{n})=r$ where $m(x)$, $n(x)$ are polynomials relatively prime…
corcia candy
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