Questions tagged [valuation-theory]

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

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Lattice and extension of scalar of vector space over valued field

Let $(K_1,w)\subseteq (K_2,v)$ be a valued field extension, where $K_1$ is a local field. Let $V$ be a finite dimension $K_1$-vector space and $L$ be a $\mathcal{O}_{K_1}$-lattice in $V$. Let $a,b\in ...
Yijun Yuan's user avatar
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Set of representatives vs Teichmüller representatives

Say $K$ is a local field (complete wrt a discrete valuation with finite residue field, or if you want perfect residue field). Denote its residue field by $k$ where $k= \mathcal{O}_K / \mathcal{M}_K.$ ...
berightback's user avatar
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What's the importance of the Approximation Theorem - Artin-Whaples Approximation Theorem

The above pictures present the statement of the approximation theorem by Artin-Whaples and its corollary in their paper. I do understand that the theorem implies that we can use one element to ...
Z Wu's user avatar
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I can't solve sections a and b in the question [closed]

Consider the AR (AutoRegressive) model, univariate: $yn​\=αyn−1​+μn​,n\=1,2,…,N$ where $y0​\=0$,$\\mu\_1, \\mu\_2, \\ldots, \\mu\_N$ are "white noises" with zero expectation and variance $\...
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What is the connection between the valuation of a polynomial of a function filed of a curve and the corresponding laurent series?

I've read somewhere an MSE that we can understand the normalized valuation of a polynomial in the function field of a curve at a smooth point as the first non-vanishing coefficient (or exponent) of a ...
Zedssad's user avatar
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1 answer
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Why is (y) a uniformizer for $y^2 = x^3 + x$ at (0,0) and why is its order equal to 1?

I'm currently trying to understand Silvermans example for the valuation on curves discussed in the answer to this post: Definition and example of "order of a function at a point of a curve" ...
Zedssad's user avatar
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Why is the valuation of a separable closure equal to $\mathbb{Q}$?

I am reading a paper by Serre, in which at some point he say We extend the valuation $v$ from $K$ to $K^{sep}$; in this way we obtain a valuation on $K^{sep}$ with value group $v(K^{sep \times})=\...
Batrachotoxin's user avatar
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Convergence of $\left \{c^{q^k}\right \}$ with finite residue field with $q$ elements

The problem Let $ν$ be a normalised discrete order function of a field $F$. Suppose that $F$ is complete. Put $$R=\left\{x\in F:v\left(x\right)\geqslant0\right\}$$ and $$ M=\left\{x\in F:v\left(x\...
Zhang's user avatar
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Prove the p-adic valuation of n! equals a floor function sum

I've seen related questions, but none quite like this one. For a rational number $r$ let $[r]$ be the largest integer less than or equal to $r$, e.g., $[\frac{1}{2}] = 0, [2] = 2$, and $[3\frac{1}{3}] ...
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associate a character to an abelian extension in local class field theory

Let $K_v$ a local and $F$ a finite abelian extension of it. Two questions: Could somebody explain how local class field theory associates naturally a character of $K_v^{\times}$ to $F$. And, if $r \in ...
JackYo's user avatar
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embed roots of unity from valued field into its residue field

Let $K_v$ non-archim valued complete local field with finite residue field $\kappa= \mathcal{O}_K / \mathfrak{m}_v$ of characteristic $p$. Assume $K_v$ contains $d$th roots of unity $U_d$. Is there ...
JackYo's user avatar
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Elliptic curves - mod $\mathcal{M}$ map

I am struggling with an example from my lecture notes. Let $K$ be a non-Archimedean field with valuation ring $R=\{x:|x|\le 1\}$, maximal ideal $\mathcal{M}=\{x:|x|<1\}$ and residue field $k=R/\...
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$(\varprojlim_n O_K/p^n) \otimes_{O_K} O_L \cong \varprojlim_n (O_K/p^n \otimes_{O_K} O_L)$

Let $L/K$ be an extension of global fields, and $O_K$, $O_L$ be respectively integer rings of $K$, $L$. Let $p$ be a prime ideal of $O_K$. When is the canonical homomorphism $$(\varprojlim_n O_K/p^n) \...
mouinow's user avatar
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A case where submodules, isomorphic as modules, are isomorphic as submodules

Let $R$ be a discrete valuation ring with maximal ideal $m$ and let $M$ be a finitely generated torsion $R$-module. Let $K$ and $K'$ be two submodules of $M$ which are isomorphic as $R$-modules. Can ...
Stabilo's user avatar
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Valuation on $K(x, y)$ similar to $\mathfrak{p}$-adic valuation

Let $K$ be a field. Since $K[x]$ is a Dedekind domain, constructing valuations for $K(x)$ is easy - we can just take any prime ideal of $K[x]$ and consider the $\mathfrak{p}$-adic valuation. So ...
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Analytic equivalence and isomorphism of valued fields

Let $(F,v)$ be a valued field, with two extensions $(E_1,w_1)$ and $(E_2,w_2)$, i.e. $E_1$ (resp. $E_2$) extends $F$ as a field and $w_1$ (resp. $w_2$) extends $v$. We call $E_1$ and $E_2$ ...
Yijun Yuan's user avatar
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Why is this proof of an inequality in a DVR valid? [duplicate]

Let $R$ be a discrete valuation ring. We have a valuation $v$ on the fraction field $K$ of $R$. Let $x,y \in K$ with $v(x)<v(y)$. I want to prove that $v(x+y) = \min(v(x), v(y))$. So, we have $v(x) ...
Mr Prof's user avatar
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Is the dimension of a valuation ring equal to the rank of its value group?

Here it is claimed that: The dimension of a valuation ring is equal to the rank of its value group. I couldn't find confirmation of this statement so I'm trying to prove it myself. Here is what I ...
Anakhand's user avatar
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What's a direct or quickest proof of simple zeros lift to factorizations lift?

Is there a direct proof that for a valued field $(F,v)$, the existence of lifts of simple roots in $k_v$ (the residue field) to $\mathcal{O}_v$ (the valuation subring) implies that factorizations in $...
Alvaro Pintado's user avatar
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Why do we fix an extension of valuation of $K$ to $\overline{K}$?

Let $K$ be a valuation field. Let $v$ be a valuation of $K$. Let $K_v$ be completion of $K$ at $v$. I often encounter an expression. Fix an extension of $v$ to $\overline{K}$, which serves to fixes ...
Pont's user avatar
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How to express elements of a DVR as power series of the uniformizer?

Let $R$ be a DVR with a fixed uniformizer $t$. My main considering example is the local ring of an algebraic curve over an algebraically closed field $k$, so let's assume $R$ is a $k$-algebra and has ...
S.Gau at Math's user avatar
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Can we define valuation(s) in any integral ring?

Let $R$ be an integral domain with unity, that is not a field, and $\mathfrak{m}$ a maximal ideal in $R$, that is not the $(0)$ ideal. I am following a course in which I learned that when the ...
Arthur-14's user avatar
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1 answer
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Value group of the $p$-adic complex numbers

I have read that the value group of the $p$-adic complex numbers ($\mathbb{C}_p$) and the algebraic closure of $\mathbb{Q}_p$ is $\mathbb{Q}$. I understand why the value group of $\mathbb{Q}_p$ is $\...
Pambra iskra's user avatar
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Valuation rings come from valuation functions

There is a theorem that states that every valuation ring of a field $k$ comes from a valuation function on $k$. In the proof, you take a valuation ring $R$ and define the valuation $v:K^{\times}\...
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Non-isomorphic models of Hen$_{0,0}$ with isomorphic RV sorts

I wonder the existence of non-isomorphic models of $\text{Hen}_{0,0}$ with isomorphic RV sorts. To clarify the notation, $\text{Hen}_{0,0}$ means Henselian valued field with equicharacteristic $(0,0)$,...
Louiseeeee's user avatar
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Prolongation valuation of non-complete valued field

Let $(K,v)$ be an algebraically closed complete valuation field. Let $(R,v|_R)$ be a valuation subring of $K$. Denote by $F^{\operatorname{sep}}$ the separable closure of $\operatorname{Frac}R$ in $K$....
Yijun Yuan's user avatar
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1 answer
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Valuation ring $(A,\mathfrak{m})$ so that $\text{Spec }A\setminus\{\mathfrak{m}\}$ has no closed points

Let $A$ be a valuation ring with maximal ideal $\mathfrak{m}$ which is the union of the non-maximal prime ideals of $A$. One can show that for $X=\text{Spec }A$, the open subscheme $X\setminus\{\...
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Showing degree of global extension is sum of degree of local extensions

I am trying to understand why, for a global extension $F \subset K$, given a place $\mu$ over $F$, we have that $$n = \sum_{\nu: \nu | \mu} n_{\nu}$$ where $n = [K:F]$, $n_{\nu} = [K_{\nu}|F_{\mu}]$ ...
algebroo's user avatar
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1 answer
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Is a field with archimedean absolute value with compact unit sphere complete?

By the unit sphere of a valued field $(K,|\cdot|)$ I mean $\{x\in K:|x|=1\}$. We know that if the unit closed ball $\{x\in K:|x|\le 1\}$ of a valued field $K$ with nontrivial absolute value is compact,...
Jianing Song's user avatar
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Completion of global field = extension of completed sub-field?

Given a global field $K = F(\alpha)$ as a finite seperable extension of $F$, where $F = \mathbb{Q} $ or $\mathbb{F}_q(t)$, suppose that $u$ is a fixed place of $F$, and that $v$ is an extension of $u$ ...
algebroo's user avatar
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1 answer
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Is there any relationship between domination of local rings and extension of ring homomorphisms to an algebraically closed field?

What is the relationship between domination of local rings and extension of ring homomorphisms to an algebraically closed field? Let $K$ be a field, $A,B$ local rings contained in $K$. We say $B$ ...
shintuku's user avatar
1 vote
1 answer
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Question about Dedekind Domains and Valuations

I've been trying to prove this claim that I think is true, but I'm getting more and more sure it's not true. Here's the set up: Let $\mathcal{O}$ be a Dedekind domain, and $K$ its field of fractions. ...
littleman's user avatar
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Does a pair of complex embeddings correspond to a single complex prime?

I'm reading through Neukirch's Algebraic Number Theory, and he distinguishes between a number field's real primes, given by its real embeddings, and its complex primes, which are induced by the pairs ...
littleman's user avatar
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Characterization of Prüfer domains via valuation rings and modules [duplicate]

I am learning valuation rings from Matsumura’s Commutative Ring Theory book. There is an exercise that characterizes Prüfer domains among integral domains. Let $A$ be an integral domain. The exercise ...
Boris's user avatar
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1 vote
1 answer
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Is every archimedean absolute value over $\mathbb{C}$ complete?

By an absolute value over $\mathbb{C}$, I mean a function $\mathbb{C}\to \mathbb{R}_{\ge 0}$ such that: $\bullet$ $|x| = 0\Longleftrightarrow x=0$; $\bullet$ for all $x,y\in\mathbb{C}$ we have $|xy| = ...
Jianing Song's user avatar
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3 answers
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Prove $v_p(a^2+b^2)$ is even for $a, b\in\mathbb{Z}_p$ and $p\equiv 3\pmod 4$.

Trying to prove that the $p$-adic valuation of $a^2 + b^2$ is even for $p$-adic integers $a$ and $b$ when $p$ is a $3\bmod 4$ prime. The result for integer $a$ and $b$ is a lemma in elementary number ...
Tim's user avatar
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Complete non-Archimedean field with infinite residual field

At the end of wolfram article on non-Archimedean valuation, it is claimed that the residual field of a complete non-Archimedean field is finite. This article does not have compactness assumptions on $...
aaa acb's user avatar
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2 answers
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Is $v((f \circ g)(x)) \geq v(f(x))+v(g(x))$, where $v$ is p-adic valuation?

Let $\mathbb Q_p$ be a $p$-adic field and $f,g \in \mathbb Q_p[x]$. Suppose $v$ is a $p$-adic valuation, and let $f \circ g$ be the formal composition. Is $v(f \circ g) \geq v(f)+v(g)$ ? For the ...
MAS's user avatar
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4 votes
1 answer
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If $|\cdot|$ is a nontrivial nonarchimedean absolute value over $\mathbb{C}$, must $(\mathbb{C},|\cdot|)$ embed into $\mathbb{C}_p$ for some $p$?

By an absolute value over $\mathbb{C}$, I mean a function $\mathbb{C}\to \mathbb{R}_{\ge 0}$ such that: $\bullet$ $|x| = 0\Longleftrightarrow x=0$; $\bullet$ for all $x,y\in\mathbb{C}$, we have $|xy| =...
Jianing Song's user avatar
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1 answer
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valuations of $\mathbb{Q}(\sqrt[3]{2})$ [closed]

I saw a questions in a book (that unfortunately I don't remember it's name!) and couldn't solve it. What are Archimedean and non-Archimedean valuations of the field $\mathbb{Q}(\sqrt[3]{2})$?
Meysam Eskandari's user avatar
1 vote
0 answers
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Is it possible for an incompletely valued field to have a finite-degree extension which is complete?

Is it possible to have a completely valued field $(L,|\cdot|)$ and a subfield $K$ of $L$ such that $\bullet$ $L/K$ is a finite extension and that $\bullet$ $(K,|\cdot| \, |_K)$ is not complete? Note ...
Jianing Song's user avatar
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1 vote
1 answer
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Prove that the 5-adic valuation of $(10^{k+t}+10^t+1)^c-1$ is $t+v_5(c)$

Here is a result of mine that I really wish to be strictly proven (I need it as an intermediate lemma for a theorem). Let $v_5(a)$ indicate the $5$-adic valuation of $a \in \mathbb{Z}^+$. How can we ...
Marco Ripà's user avatar
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1 vote
1 answer
96 views

Does $(3+2\sqrt{-2})y^2=1+12\sqrt{-2}-4・17x^4$ have solution in $\Bbb{Q}_{17}$?

Does $(3+2\sqrt{-2})y^2=1+12\sqrt{-2}-4・17x^4$ have solution in $\Bbb{Q}_{17}$ ? I feel this question is very difficult. Here, $\sqrt{-2}$ denotes one of root of $x^2+2=0$ in $\overline{\Bbb{Q}_{17}}$ ...
Pont's user avatar
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What is $17$ adic value of $3+2\sqrt{-2}\in \Bbb{Q}_{17}$?

What is $17$ adic value of $3+2\sqrt{-2}\in \Bbb{Q}_{17}$ ? I expanded $\sqrt{-2}=7+a_117+・・・$, so $3+2\sqrt{-2}=17+{a_1}17+・・・=17(1+a_1+a_{2}17+・・・)$, thus $ord_{17}(3+2\sqrt{-2})\ge 1$. If $ord_{17}(...
Pont's user avatar
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1 vote
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does the residue field of an algebraically closed valued field lift?

Let $v:K\to \Gamma$ be a valuation on a field $K$. I know that if the valuation ring $\mathcal O_v$ is henselian and $k$ has characteristic zero, then $k$ has a lift to $K$. I.e. there is a section $s:...
ugur efem's user avatar
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0 answers
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Why's $v_L (x)=\frac{1}{n}v_K(N_{L/K}(x))$ integer for unramified local field extension $L/K$?

If $K$ is a local field and $L/K$ is a finite extension, then the valuation $v_K$ can be extended uniquely to a valuation $v_L$ of $L$ such that $v_L$ restricted to $K$ is equal to $v_K$. This is one ...
Pont's user avatar
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0 votes
1 answer
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Unramified extension in CDVF

Let $(K,v)$ a complete discrete valuation field of mixed characteristics $(0,p)$ and let $\bar{K}$ ann algebraic closure of $K$. We can define an additive valuation $w: \bar{K} \to \mathbb{Q}\cup \{\...
Mario's user avatar
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3 votes
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Why is the generalized definition of valued field automatically non-Archimedean?

The classical definition of a field $K$ with an absolute value $|\cdot|:K\to \mathbb{R}_{\geq 0}$ is that $\forall x,y\in K$ $x=0\Leftrightarrow|x|=0$ $|xy|=|x|\cdot|y|$ $|x+y|\leq |x|+|y|$ If the ...
Z Wu's user avatar
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0 votes
2 answers
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Need of valuation theory

In many books, valuations are introduced with the motivation of generalization of divisibility. But, I could not touch it and digress, due to simple questions from elementary concepts in algebra. If $...
Maths Rahul's user avatar
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2 votes
1 answer
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Subfield of a complete valuation field also complete

If we have a complete field $K$ with respect to a discrete valuation $v$ and $L\subset K$ is a finite separable subfield, is $L$ complete with respect to the restriction of the valuation $v$?
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