For questions related to valuation functions on a field.

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Isomorphism of completions of number fields

Let $K$ and $L$ be number fields, $v$ a place of $K$ (either archimedean or non-archimedean) and $\theta:K\simeq L$ a ring isomorphism. I am trying to show that $\theta$ induces an isomorphism ...
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435 views

An application of the Weak Approximation theorem - Artin-Whaples Approximation Theorem

Let us recall the weak approximation theorem from Valuation theory in Algebraic Number Theory. Let $K$ be a field, and let $|\cdot|_{1},\dots, |\cdot|_n$ be pairwise non-equivalent nontrivial ...
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Determine all discrete valuations on $\mathbb{C}(x)$.

To clarify, for a field $K$, a valuation $v$ on $K$ is a map $v:K^{\times}\to G$ for $G$ an ordered group (written additively) such that for any $a,b\in K^{\times}$: 1) $v(ab)=v(a)+v(b)$; 2) ...
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The Archimedean place of $\mathbb{Q}$

Is there a way to extract the Archimedean absolute value of $\mathbb{Q}$ from its field structure in a way analogous to its non-archimedean absolute values? Here is some context: Given a valuation ...
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Does a place $v$ of a number field $K$ ramify in $L/K$ iff $v\mid d_L$?

Let $L/K$ be an extension of number fields and $v$ be a prime (an equivalence class of valuations) of $K$ and $d_L$ the absolute discriminant of $L$. I know that a rational prime $p$ in $\mathbb Q$ ...
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355 views

Valuation rings

What's the spectrum of a valuation ring? How to describe morphisms from it to a scheme? Is it enough to set the image of generic point and of a maximal ideal and correspondent map of local rings?
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non-Archimedean Valued field extension of $\mathbb{R}$

Let $K$ be a field with non-Archimedean valuation $|\cdot|$. Suppose that $\mathbb{R}\subset K$. Question 1: Is the restriction of $|\cdot|$ to $\mathbb{R}$ the trivial valuation? I guess that the ...
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Proving that $\mathbb{Q}_p$ is not formally real.

I've been looking for a concrete proof without results. The only hint that I have found says: 1) $\mathbb{Q}_2$ contains a square root of $-7$. 2) $\mathbb{Q}_p$ ($p>2$) contains a square root of ...
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Valuation rings of dim 1,2

I am studying valuation rings (beginner). I have read some theorems but still don't know a nontrivial example. Please give me an example which is not field. Also Need help to have examples of Krull ...
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1answer
24 views

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 ...
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35 views

$p$-adic field with infinite residue field. [duplicate]

I am reading J M Fontaine's book where on page 7 the following definition is made: A local field($K$) is a complete discrete valuation field whose reside field($k$) is a perfect field of ...
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45 views

Majoration of the $p$-adic valuation of a factorial.

Let $p$ be a prime number. In order to prove a result on $p$-adic interpolation of iterates, I need to show the following: Lemma. Let $m$ be an integer, one has: ...
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Principal local Artinian ring is a quotient of discrete valuation ring.

I have seen here the following statement: Let $R$ be a principal local Artinian ring. Clearly the quotient of a discrete valuation ring is such a ring; conversely it is not difficult to show that ...
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Is the result true when the valuation is trivial and $\dim(X)=n$?

Here I proved the following result: Proposition: Let $(K,|\cdot|)$ be a valued field with non-trivial valuation and let $X$ be a vector space over $(K,|\cdot|)$. Two norms $p_1,p_2$ on $X$ are ...
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Is the result true when the valuation is trivial?

Here I proved the following result: Proposition: Let $(K,|\cdot|)$ be a valued field with non-trivial valuation and let $X$ be a vector space over $(K,|\cdot|)$. Two norms $p_1,p_2$ on $X$ are ...
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1answer
53 views

Is true the following statement?

Consider $0<s<1$. Is true the following statement? $(\forall 0<\delta<1)(\exists \gamma>0)$ such that $(\forall x>0)(\exists n\in\mathbb{Z})$ such that $\gamma<s^nx<\delta. ...
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2answers
57 views

Discrete valuation ring and finitely generated submodules

Let $R$ be a Discrete Valuation Ring with fraction field $K$. Will this imply any proper $R$-submodule of $K$ is finitely generated (hence a fractional ideal)? I know $K$ is not finitely ...
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Discrete Valuation Rings with property that $v(x+y)=\min(v(x),v(y))$ [duplicate]

Let $R$ be a discrete valuation ring on a fraction field $K$ of $R$. If $x,y\in K$ such that $v(x)< v(y)$, prove that $v(x+y)=\min(v(x),v(y))$. By definition of discrete valuation, we have that ...
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1answer
28 views

Valuation ring and integral closure

Let $A$ be a one-dimension local noetherian domain and suppose that we know that $K=\text{Frac}(A)$ is a complete discrete valuation field (valuations for me are surjective). Let's denote with ...
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1answer
20 views

Local subring of a DVR and finite residue field extension

Let $\mathcal O$ be a complete DVR with fraction field $K$, maximal ideal $\mathfrak p$ and residue field $\widetilde K=\mathcal O/\mathfrak p$. Now consider a subring $A\subset \mathcal O$ with the ...
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1answer
51 views

Localization of a valuation ring at a prime is abstractly isomorphic to the original ring

Let $A$ be an integral domain with quotient field $K$. We say that $A$ is a valuation ring if for any $0 \neq x \in K$, $x$ or $\frac{1}{x}$ lies in $A$. Then $A$ is necessarily a local ring. If ...
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Can a valuation ring properly contains another valuation ring with the same field of fractions?

Definition of valuation ring: Let $R$ be an integral domain with $frac(R)=K$. Then $R$ is said to be a valuation ring if (1) $R \neq K$ (2) $\forall x \in K, x \in R$ or $x^{-1} \in R$. Now my ...
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1answer
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The torsion subgroup of principal units $U^{(1)}$

$\newcommand{\U}{U^{(1)}}$ $\newcommand{\O}{\mathcal{O}}$ $\newcommand{\p}{\mathfrak{p}}$ $\DeclareMathOperator{\char}{char}$ $\newcommand{\N}{\mathbb{N}}$ I have a question about the torsion ...
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45 views

Extension of scalars and completions

Suppose that $A$ is a Noetherian regular (added later) local domain. Moreover $\widehat A$ is $\mathfrak m$-adic completion $\widehat A$ w.r.t the maximal ideal and $K$ is the fraction field of $A$. ...
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Absolute values on $\mathbb{R}$ [closed]

Two related questions: a) is there some characterization of all the absolute values on $\mathbb{R}$? (similar to Ostrowski for $\mathbb{Q}$) b) are there non Archimedean absolute values on ...
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43 views

Is $D=\left \{ x\in\mathbb{R}: \left | x \right |\leq 1 \right \}$ a discrete valuation ring?

A discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. Is $D=\left \{ x\in\mathbb{R}: \left | x \right |\leq 1 \right \}$ a discrete valuation ...
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1answer
48 views

There are no archimedean function fields

Definition: a field $L\supseteq K$ is called a function field over $K$ if the extension $L|K$ is finitely generated, regular and of transcendence degree $1$. In the book "Topics in the theory of ...
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extension of valuation rings

Assume $L$ is a finite Galois extension of $K$, and $R$ is a valuation ring on $K$ with maximal ideal $\mathfrak{m}$. Is it true that there are only finitely many valuation rings $(O,\mathcal{M})$ ...
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1answer
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If $v$ is a valuation and $v(x)<v(y)$ then $v(x+y)=v(y)$. [duplicate]

I was studying from some book and I came across something I haven't been able to justify. Let's suppose se have a field $K$ and an ordered group $G$ (with multiplicative notation) with an extra ...
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extension of a valuation on a function field

Let $K$ be a field, and $K(x)$ the field of rational functions over $K$. Consider the degree valuation $v$ on $K(x)$, That is $v\left(\frac{f(x)}{g(x)}\right)=\deg(g)-\deg(f)$. So for every $f(x)\in ...
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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 ...
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1answer
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A question concerning archimedean places on number fields

Let $L$ be a number field and $G=Aut(L\mid \mathbb{Q})$. Let $|\cdot|$ be the usual archimedean value on $\mathbb{C}$ and, by abuse of notation, its restriction to $\mathbb{Q}$. Then the archimedean ...
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1answer
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Valuation ring between $F$ and $\mathcal O_F$

Let $(F,v)$ be a complete discrete valuation field (normalized) with ring of integers $\mathcal O_F$. Why cannot exist a valuation ring $A$ of $F$ such that $F\supsetneq A\supsetneq\mathcal O_F$ ...
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1answer
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Does every non-archimedean absolute value on field take value in $\mathbb{Q}$

Let $K$ be a field, a non-archimedean absolute value is defined to be a map $K\to \mathbb{R}$ satisfying $|x|=0\Rightarrow x=0$, $|x|\cdot|y|=|xy|$ and $|x+y|\leq\max(|x|,|y|)$. Is there an example ...
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1answer
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Restricted valuation on subring of a DVR

Let $\mathcal O$ be a normalized discrete valuation ring. This means that there is a surjective valuation: $$v:\text{Frac}\,(\mathcal O)\rightarrow\mathbb Z\cup \left\{\infty\right\}.$$ Now consider ...
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Extending $2$-adic valuation to a finite extension of $\mathbb Q$.

Say we want to extend the $2$-adic valuation of $\mathbb Q$ to $\mathbb Q[x]/x^3-3$. First, we determine the possible valuations of $x$. Since $x^3=3$ and since $3$ has $2$-adic valuation $0$, $x$ ...
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Function field on a non-singular algebraic curve as the field of meromorphic functions

Let $k$ be an algebraically closed field and let $X$ be a non-singular algebraic projective curve over $k$. Very often, when most books present the function field $k(X)$, they say that "it is the ...
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1answer
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All local subrings of $\mathbb Q$ are valuation rings of $\mathbb Q$?

Let $R\subseteq\mathbb{Q}$ be a local subring with maximal ideal $\mathfrak{m}$. Is $R$ a valuation ring of $\mathbb Q$? $R$ is a valuation ring iff its ideals are linearly ordered. But I'm stuck ...
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1answer
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Ramification of primes in number fields in terms of valuations

Let $L/K$ be a number field extension. Is there a definition of ramification of primes(both infinite and finite) in terms of the valuations induced? This answer gives a definition for infinite ...
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How to write the Adeles over $\mathbb{Q}(i)$?

What do these adeles look like over $\mathbb{Q}(i)$? These are simply fractions where the numerator and denominator are allowed to have the number $i = \sqrt{-1}$. The elements look like: $$ ...
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how can I prove this statement in local field theory?

Let $\mathbb{K}$ be a fixed local field. Then there is an integer $q=p^r$, where p is a fixed prime element of $\mathbb{K}$ and r is a positive integer [edit] such that for each $x\in\mathbb{K}, ...
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A construction in the proof of “any local ring is dominated by a DVR”

Let $O$ be a noetherian local domain with maximal ideal $m$. I want to prove: for a suitable choice of generators $x_1,\dots,x_n$ of $m$, the ideal $(x_1)$ in $O'=O[x_2/x_1,\dots,x_n/x_1]$ is not ...
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Is every integral domain contained in a discrete valuation ring?

Is is true that every integral domain which is not a field is contained in a proper subring of its fraction field which is a DVR?
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Compare Valuations

Let $K$ be a field complete with respect to a discrete valuation $v$. Let $K_s$ its separable closure, $w$ the unique valuation extending $v$, $(\mathcal{O}_{K_s}$, $\mathfrak{M})$ respectively the ...
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1answer
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Ramification group definition in Neukirch's Chap. 2 Section 9

Let $L/K$ be an algebraic Galois extension and $w/v$ a non-archimedean extension of valuations. Let $\mathcal{O}$ and $\mathfrak{P}$ denote the valuation ring and valuation ideal of $(L,w)$. After ...
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1answer
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Property of valuation in $\Bbb{Z}_p$

Let $v$ denote the $p$-adic valuation in $\Bbb{Z}_p$. Let $a_1,a_2,a_3$ be three elements of $\Bbb{Z}_p$. Then I want to show that $$\min \{ v(a_1), v(a_2), v(a_1+a_2+a_3)\}=\min ...
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1answer
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Topology for the decomposition of $L\otimes_K K_\frak p$

Let $L/K$ be an extension of number fields, $\frak P|\frak p$ primes of $\mathcal O_L$ and $\mathcal O_K$, and let $K_\frak p$ be the completion of $K$ wrt $|\cdot|_\frak p$ and $L_\frak P$ the ...
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2answers
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Why does discrete valuation need to be surjective?

My question appeared while I was solving a problem in Abstract Algebra of Dummit and Foote. That is problem 26. My solution is almost the same as the proof in this link. However, I'm still ...
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finite extensions of discrete valuation fields: A method to find a basis

Suppose that $L|K$ is a finite extension of discrete valuation fields. Namely $w$ is a discrete valuation on $L$ extending a valuation $v$ on $K$. Now consider the respective rings of integers ...
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Minimal polynomial has coefficients in the valuation ring?

Question Let $K$ be a complete nonarchimedean valued field with valuation ring $\mathcal{O}$ and let $L/K$ be a finite extension. Let $\alpha$ be an element of $L$ and $f\in K[x]$ its minimal ...