# Questions tagged [valuation-theory]

For questions related to valuation functions on a field.

430 questions
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### About the concept of a valuation ring

I got a little confused by the different definitions of valuation rings while reading Atiyah's introduction to commutative algebra. Let $A$ be an integral domain and $K$ its field of fractions. We ...
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### Norms, Orders, and 'almost' valuations in Number Fields

This is one of those questions that feels true, but I can't prove or disprove it. Let $K = \mathbb{Q}(\sqrt{D})$, where $D < 0$ is squarefree, and let $k>0$ be some positive integer. If you ...
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### Definition of the Norm Residue Symbol

Let $K$ be a field and $v$ its valuation. Let $K_v$ denotes the completion of $K$ with respect to the valuation $v$. If $L/K$ is a finite Galois extension, then all the $L_w$, for $w$ extending $v$, ...
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### Units in a discrete valuation ring.

I'm doing problem 2.26 in the book "algebraic curves" by Fulton: http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf One is given two DVR's R and S both of which have the same field of fractions $K$...
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### Smallest infinite place of a rational function field

Let $f$ be a separable irreducible polynomial of degree $n$ with $f \in F[x]$, where $F$ is a function field. I try to understand the following part of an article: ..., we use the minimum infinite ...
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### Can a given field have multiple value groups?

Suppose $v : K \rightarrow Γ$ is a surjective valuation, where $K$ is the field and $Γ$ is the value group. The set $A = \{x : v(x) \geq 0\}$ is a valuation ring and $U=\{x :v(x)=0\}$ are the units of ...
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### Why is the residue field of $\mathbb{Q}_p$ isomorphic to $\mathbb{F}_p$?

$\mathcal{O}=\{x\in \mathbb{Q}_p:v(x)\geq0\}$ is a valuation ring. $\mathfrak{m}=\{x\in \mathbb{Q}_p: v(x)>0\}$ is the maximal ideal of $\mathcal{O}$. Why is $K=\mathcal{O}/\mathfrak{m}$ ...
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### Relation between tame symbol and residue on a curve

For an discrete valuation field $K$ we can define the tame symbol: $$(\,,\,)_K:K^\times\times K^\times\to \overline K^\times$$ $$(a,b)\mapsto(-1)^{v(a)v(b)}\overline{a^{v(b)}b^{-v(a)}}$$ Consider ...
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### Trace map for extension of local fields

Let $K\supset F$ a finite extension of local fields. It means that the valuation $v_K$ extends the valuation $v_F$. We denote with $\pi_K$ and $\pi_F$ the uniformizer parameters and with $\mathcal O_K$...
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### Extending absolute values on local fields - what is the 'correct' normalization and the relation to the global theory?

I'm having a tough time figuring out the 'correct' normalization for extending absolute values of local fields. I'm also trying to piece together how this interacts with the global theory, so below is ...
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### Given a valuation on a field , does it always extend to a valuation on every extension field

Let $(G,+,\ge)$ be a totally ordered abelian group. Let $K\subseteq L$ be an extension of fields. Let $v : K\setminus \{0\} \to G$ be a valuation (https://en.wikipedia.org/wiki/Valuation_(algebra)) . ...
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### Desingularisation of curves (Lorenzini-Invitiation to Arithmetic Geometry, chap 6,ex 7)

Given a nonsingular complete curve over algebraically closed $\bar{k}$, which is interpreted as a field $\bar{k}(X)$ of transcendence degree 1 and its set of valuations trivial on $\bar{k}$, we may ...
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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? ...
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### Galois Groups in Ramification Theory

I had a slight confusion about Galois groups over a base field which is complete with respect to a discrete valuation. We know that there are irreducible polynomials such as $X^3+X^2+2X-8$ where ...
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### non commuative division ring with a discrete valuation

I do not manage to exhib a non commutative ring of division with a discrete valuation. Can anyone show me one? Examples with quaternions would be a plus !! Thanks in advance.
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### How to find the Newton polygon of the polynomial product $\ \prod_{i=1}^{p^2} (1-iX)$

How to find the Newton polygon of the polynomial product $\ \prod_{i=1}^{p^2} (1-iX)$ ? Answer: Let $\ f(X)=\prod_{i=1}^{p^2} (1-iX)=(1-X)(1-2X) \cdots (1-pX) \cdots (1-p^2X).$ If I multiply , ...
1answer
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### Ideals in valuation rings [closed]

How do I prove the fact that for any valuation ring $V$ the ideals are totally ordered under inclusion?
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### Valuation ring's principal ideals [closed]

Let $V$ be a valuation ring. Then any two principal ideals $A_1$ and $A_2$ of $V$ are ordered by inclusion. I need a proof for this lemma and I don't know how to start.
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### On the maximal unramified extension of $\mathbb{Q}_{p}$ of a given degree.

I'm stucked with a theorem withouth proof saw in a Book about G-Functions by Dwork, I will appreciate any hint, also I provide a ''proof'' of that theorem, but a feel that is too ''bla bla'' and I ...