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Questions tagged [valuation-theory]

For questions related to valuation functions on a field.

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Completing Algebraic Integers into Squares

Let $L/K$ be an extension of number fields with Galois closure $E$, and let $\theta \in \mathcal{O}_L \setminus \{0\}$. Let $\Sigma_E$ be the set of primes of $E$, let $S' \subset \Sigma_E$ be a ...
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43 views

Bound for roots of a polynomial with coefficients in a non-Archimedean valued field

Is there any bound for the valuation of the roots of a given polynomial with coefficients in an algebraically closed non-Archimedean valued field? Any reference or insight would be appreciated.
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95 views

Fraction field of $A[[t]]$

Let $A$ be a complete discrete valuation ring. (For example $A$ is the completion of the local ring of a curve at a point). What is the fraction field of $A[[t]]$? Is it $\operatorname{Frac}(A)((t))$?...
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40 views

Valuation of a number field element over a prime ideal in an order

Having read https://mathoverflow.net/questions/144671/number-field-sieve-for-factorization-with-non-monic-non-linear-polynomial-cant I stumbled on a problem I can't prove. Most of the questions posed ...
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1answer
63 views

Valuation ring, of infinite global dimension, with principal maximal ideal

Does there exist a Valuation ring $(R, \mathfrak m)$ , with principal maximal ideal, of infinite global dimension ? Corollary 2 of the following paper by Osofsky has an example of Valuation ring of ...
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2answers
30 views

good reference for studying valuations theory (algebra)

I need some good reference for studying valuations, I was looking for some really detailed material, as I'm just starting to study this theme. Please help.
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27 views

On the first and second Ext modules over some valuation rings

Let $(R, \mathfrak m)$ be a valuation ring of finite Krull dimension, with non-principal maximal ideal. So that $\mathfrak m^2=\mathfrak m$. If $M$ is an $R$-module with $Supp M=\{ \mathfrak m \}$ , ...
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1answer
38 views

Complete valuation, norm of finite extension. Proof of Propositon.

I would like to ask for tips how to manage with proof of this proposition. How to show that $v$ can be uniquely extended to $v'$ ? Should I assume that $v$ can be also extended to another valuation? ...
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1answer
56 views

Proving that “Giving a DVR with quotient field K is equivalent to defining an order function on K”

I'm trying to solve problem 2.28 in Fulton's Algebraic Curves. The problem is the following: An order function on a field $K$ is a function $\phi:K\to \mathbb{Z} \cup {\{\infty}\}$ satisfying: i) $\...
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1answer
64 views

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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22 views

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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25 views

Clarification on a proof about $p$-valued groups.

First we have a few definitions: $(R,v)$ is a filtered ring if $v : R \rightarrow \mathbb R^{\geq0} \cup \{\infty\}$ satisfies: $v(r-s) \geq \min\{v(r), v(s)\}$ $v(rs) \geq v(r) + v(...
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Are Valuations on algebraic extensions of an henselian field unique?

The definition of henselian is: A valued field $(\mathbb{K},\nu)$ is said to be Henselian if for any algebraic extension $\mathbb{L}$ of $\mathbb{K}$ there is a unique valuation $\tilde{\nu}$ on $\...
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1answer
52 views

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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1answer
61 views

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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26 views

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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1answer
92 views

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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1answer
31 views

Valuation of $x^\lambda$ in a complete, $p$-valued group

Suppose for $p$ a prime that $(G,\omega)$ is a complete $p$-valued group, $x \in G$ and $\lambda \in \mathbb Z_p$ (the $p$-adic integers). Let $x^\lambda$ denote the unique element of $G$ such that $\...
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1answer
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The residue field of valuations are finite extension

Let $K\mid k$ be a finitely generated extension (maybe of transcendence degree bigger than one) and consider a rank one valuation of $K$ over $k$, that is, a function $$v:K^\times\rightarrow \mathbb{R}...
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1answer
26 views

Explicit example of complete discrete valuation field with prescribed residue field

Let's fix the field $\overline F:=\mathbb F_p(t)$. What is the explicit expression of a complete discrete valuation field $F$ of characteristic $0$ which has $\overline F$ as residue field? Note that ...
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The valuation attached to a smooth point of an algebraic variety

I am interested in proving the following result: Let $k$ be an algebraically closed field, $X$ a normal integral variety over $k$ and $x\in X$ a closed point. Write $\mathfrak{m}_x$ for the ...
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On the Newton Polygon for $p-$adic Power series

I'm studyng a Book about $p-$adic numbers, and I have troubles with a "degenerate" case of a Newton polygon. Let $f(X)=\sum a_{i}X^{i}\in\mathbb{Q}_{p}[\![X]\!]$, we define the Newton poligon of $f$ ...
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2answers
51 views

$p$-adic power series and its maximum in the unit ball

Let $\mathbb{K}$ be an algebraically closed field with a complete absolute value and denote by $R$ its valuation ring. Consider a power series $$f(X)=\sum_J a_JX^J\in \mathbb{k}[[X_1,\dots,X_n]]$$ ...
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55 views

On $A$ algebra homomorphisms $A[[X_1,…,X_n]]\to Q(A)$, where $A$ is a complete DVR

Let $(A,\mathfrak m)$ be a complete Discrete Valuation Ring (complete w.r.t. the $\mathfrak m $-adic topology) with fraction field $K$. Let $\phi : A[[X_1,...,X_n]]\to K$ be an $A$-algebra ...
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Extension of valuation in iterated Laurent series fields

according to well known theorems, every valuation of a field $K$ can be extended to a valuation of a field extension $L$ of $K$ and this can be explicit in the case of finite extensions. My question ...
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1answer
28 views

definition of valuation ring of a place on wikipedia

In the page Valuation in Wikipedia, https://en.wikipedia.org/wiki/Valuation_(algebra) in the section "associated objects" it is written that we can associate to a valuation $v: K\to \mathbb{R}\cup \{\...
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1answer
37 views

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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1answer
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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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1answer
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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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1answer
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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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1answer
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Valuation of the p adic logarithm

I'm stuck in some propertie about the $p$-adic logarithm. The propertie comes from a proposition in a Book by Dwork which I'm studying. The proposition says: If $v_{p}(x)>\frac{1}{p-1}$, then $v_{...
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Local fields. $p$-field $(k,v)$ Proof of lemma.

I study a lemma from 'Local Class Field Theory' and I have difficulties. To understand this properly we need another lemma Lemma:1 Let $\mathfrak{l=o/p}=\mathbb{F}_q$ for a $p$-field $(k,v)$. ...
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1answer
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Valuation ring and localizations [closed]

Theorem. Let $D$ be an integral domain with identity. The following conditions are equivalent. (1) $D_P$ is a valuation ring for each proper prime $P$ in $D$. (2) $D_M$ is a valuation ring for each ...
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every ordered field $K$ has a natural valuation $v$, whose residue field is an archimedean ordered field.

Natural valuation has a convex valuation ring, in fact the valuation ring is the convex hull of $\mathbb{Z}$. How to prove that every ordered field $K$ has a natural valuation $v$, whose residue field ...
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1answer
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Local fields. A $p$ - field $(k,v)$ with $\mathfrak{l=o/p}=\mathbb{F}_q$. Proof of lemma.

$(k,v)$ i a local field, $\mathfrak{p} = \{x | x \in k, v(x) > 0 \}$, $\mathfrak{o}=\{x | x\in k, v(x) \geq 0\}$. I'm working on Local Fields and I don't understand few things in proof of this ...
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Adjoining an element with given minimal polynomial to a DVR of characteristic p.

Let $A$ be a DVR of characteristic $p$, with $\pi$ a uniformising parameter, with $K=frac(A)$ the field of fractions. Consider the extension $L=K(\alpha)$ where $\alpha$ has minimal polynomial $y^p+\...
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2answers
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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 , ...
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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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(Sanity check) the adic spectrum $\operatorname{Spa}(\mathbb{Z}, \mathbb{Z})$

I am following Scholze's and Weinstein's notes on $p$-adic geometry on http://www.math.uni-bonn.de/people/scholze/Berkeley.pdf, example 2.3.6 (page 18 as of this moment bar any updates). $\...
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1answer
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About the existence of tamely ramified extensions

I'm trying to understand the proof of the existence of tamely ramified extensions. For this, the theorem from my book says: Let $K$ be a complete field with respect to a discrete valuation, and let $\...
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0answers
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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 ...
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1answer
52 views

Is the value group of an algebraically closed valued field divisible?

Is the value group of an algebraically closed valued field divisible? Since Every existentially closed abelian group is divisible, I'm trying to show the value group is existentially closed but I don'...
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A 2-dimensional valuation domain

Let $p$ be a prime number and $T$ be an indeterminate over $\mathbb{Q}$, and set $A:= \mathbb{Z}_{(p)}$(the localization of $\mathbb{Z}$ at $p$), $B:=\mathbb{Q}[[x]]$(power series over $\mathbb{Q}$). ...
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1answer
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Is the residue field of an algebraically closed valued field algebraically closed?

Is the residue field of an algebraically closed valued field $K$ with valuation ring $A$, algebraically closed? If I take a polynomial $f(x)$ of $k_A$ since $K$ is algebraically closed it has a root ...
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37 views

Valuation ring satisfying either a.c.c. or d.c.c. on prime ideals

Let $R$ be a Valuation ring (https://en.wikipedia.org/wiki/Valuation_ring) . 1) If $R$ satisfies a.c.c. on prime ideals, then does $R$ have finite Krull dimension ? 2) If $R$ satisfies d.c.c. on ...
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2answers
139 views

Computing quotient by dividing formally in $p$-adic number system

From Gouvea's p-adic Numbers: To pass from positive integers to positive rationals, we simply do exactly as in the other case, that is we expand both numerator and denominator in powers of p, ...
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Do elements having any $p$ power roots form a open subset in a perfectoid field

Let $K$ be a perfectoid field, if $K$ has positive characterestic then every $a \in K$ has any $p$ power roots i.e $a=x^{p^n}$ have solutions for any positive integer $n$. However, in characteristic ...