For questions related to valuation functions on a field.

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1answer
44 views

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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0answers
23 views

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 ...
2
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1answer
26 views

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 ...
2
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1answer
31 views

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
30 views

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 ...
2
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1answer
21 views

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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2answers
44 views

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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2answers
50 views

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 ...
0
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1answer
34 views

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 ...
3
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1answer
51 views

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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0answers
28 views

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 ...
3
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0answers
35 views

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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0answers
42 views

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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2answers
41 views

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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0answers
24 views

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
19 views

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
29 views

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 ...
1
vote
1answer
29 views

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 ...
2
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2answers
41 views

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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2answers
30 views

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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2answers
54 views

Understanding proof that $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$ is unramified for $(n,p)=1$.

Problem Consider the extension $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$, where $\zeta$ is a $n$-th primitive root of unity and $(n,p)=1$. I want to show that $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$ is ...
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2answers
49 views

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 ...
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1answer
21 views

When do we have equality in the third axiom of valuations?

Let $R$ be a ring and $v: R \to \mathbb Z \cup \{+\infty\}$ a map that meets following conditions: $v(a) = +\infty \iff a=0$ $v(ab) = v(a)+v(b)$ $v(a+b) \geq \min\{v(a),v(b)\}$ I have to show that ...
0
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1answer
17 views

An extension $w$ of a valuation $v$ induced from $\mathfrak{p}$ come from ideal $\mathfrak{q}$ above $\mathfrak{p}$?

Assume the standard context for extension of valuations. An extension $w$ of a valuation $v$ induced from a prime ideal $\mathfrak{p}$ comes from a prime ideal $\mathfrak{q}$ above $\mathfrak{p}$? ...
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0answers
47 views

Valuations above $\mathbb{Q}(\alpha)$

Let $K$ be a field, we say that a function $\nu:K\to \mathbb{Z}_{\geq 0} \cup \lbrace \infty \rbrace$ is a valuation above $K$ if then followings hold for each choose of $x,y \in K$: 1 $\nu(x) = ...
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1answer
35 views

Separable extensions of complete valued fields are automatically totally ramified?

I come up with the following argument which seems to be too good to be true: Suppose that $L|K$ is finite separable extension of complete valued fields. Let $\nu, \nu'$ be the valuation on $L$ and ...
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1answer
29 views

Separability of complete value fields and residue class fields

Let $L|K$ be a finite separable extension of fields complete under some valuation and let $\lambda, \kappa$ be residue class fields of $L$ and $K$ respectively. I guess that we do not know ...
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0answers
50 views

Rational points of an open subset of a group scheme

Let $K$ be an algebraically closed valued field with valuation ring $\mathcal{O}$ (assume the valuation is non-trivial). Let $G$ be a group scheme over $\mathcal{O}$, $g\in G(\mathcal{O})$ and ...
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1answer
28 views

Relationship between residue class fields between extension

Let $K$ be a field with respect to a valuation and $L$ be a finite extension. For simplicity, assume $K$ is complete. From theory, valuation on $K$ extends to $L$ and the extended valuation on $L$ is ...
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1answer
15 views

Proving that certain limits exist

Let $K$ be a discrete valuation field where $\nu:K\longrightarrow\mathbb Z$ is a surjective valuation. Let $\lambda\in]0,1[\subseteq\mathbb R $, then the valuation $\nu$ induces a metric $d_\nu$ on ...
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0answers
14 views

question on proof in Frohlich/Taylor regarding extension of a discrete absolute value on a complete field

I am having some trouble understanding a proof in Fröhlich and Taylor, pages 105-106. There $K$ is a complete field with a discrete absolute value $u$ $\mathfrak o,\mathfrak p$ is the valuation ring, ...
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0answers
21 views

Characterization of valuation domains by means of their maximal ideal

I know the following theorem. Let $V, W$ be two valuation domains with the same fraction field $K$. Suppose $V$ and $W$ share the same maximal ideal. Then $V=W$. In other words, fixed a field ...
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0answers
50 views

Approach to build multivariate ranking system?

I have data of various sellers on ecommerce platform. I am trying to compute seller ranking score based on various features, such as 1] Order fulfillment rates [numeric] 2] Order cancel rate ...
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1answer
23 views

Extending valuations and linear disjointness of fields

Let $F$ be a field and let $K$ be a field extension of $F$. Suppose that $K$ can be written as the compositum of field extensions $E$ and $L$ of $F$, linearily disjoint over $F$, thus $K$ can be ...
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2answers
62 views

Reduction map on torsion points of an elliptic curve and their valuation

Let $K$ be a field of characteristic zero, complete with respect a discrete valuation $v$. Assume that the residue field $k$ is of positive characteristic $p$. Now take an elliptic curve $E$ defined ...
2
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1answer
45 views

Quotients in a non-discrete valuation ring

Let $R$ be a valuation ring, $\mathfrak{m}$ the maximal ideal of $R$. Let $k$ be the residue field, and $K$ the field of fractions of $R$. Assume that the valuation on $K$ is such that ...
3
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1answer
71 views

roots of unity in the maximal unramified extension and Kummer extensions

Let $K$ be a field complete with respect to a discrete valuation $v$ with residue field $k$ of positive characteristic $p$. Consider $$ K \subset K_{ur} \subset K_s$$ with $K_s$ the separable ...
2
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1answer
60 views

Associate a discrete valuation ring to a field $k$.

I have a field $k$ of positive characteristic $p$, not necessarily perfect. Can i find a discrete valuation ring that have $k$ as residue field and field of fractions $K$ of characteristic zero? I ...
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0answers
42 views

Motivation behind the Archimedean norm on number fields .

It is easy to justify the use of non-archimedean norms on number fields from an "inside view" as arising from the prime ideals and are therefore clearly useful a priori. However, it seems to me that ...
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1answer
20 views

Simple extensions of local fields

Let $L/K/\mathbb Q_p$ be finite extensions of local fields and let $v_L$ and $v_K$ be normalised discrete valuations on $L$ and $K$ respectively. My question is quite a general one: If the ...
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1answer
23 views

Maximal ideal of valuation

Let $\nu : K \rightarrow G \cup \{\infty\}$ be a map defined by, where $G$ is a totally ordered group and $g < \infty$ for all $g\in G.$ $\nu(a) = \infty$ if and only if $a = 0,$ $\nu(a + b) ...
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1answer
35 views

Two dimensional valuation domain with value group $\Bbb Z \oplus \Bbb Q$

Let $R$ be a two-dimensional valuation domain with prime ideals $0 \subset P \subset M$ and value group $G=\Bbb Z \oplus \Bbb Q$. Then $M^2=M$ and $P^2 \neq P$. Can we say $P^n \neq P^{n+1}$ ...
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0answers
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Projective general linear group on discrete valuation ring

Let $R$ be a complete discrete valuation ring and $k$ its residue field. Let $H$ be a finite subgroup of $PGL_2(k)$ such that its order is prime with char($k$). Is there some elementary way to show ...
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0answers
64 views

Valuations on $\Bbb Q(t)$

Ex. $2.3.3$ in Algebraic Number Theory by Neukirch is the following: Let $k$ be a field and $K = k(t)$ the function field in one variable. Show that the valuations $v_{\mathfrak p} $ associated to ...
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1answer
99 views

About the usage of the strong approximation theorem

I'm reading Henning Stichtenoth's Algebraic Function Fields and Codes and at Proposition 3.2.5(a) of section 2, chapter 3 he says: Let $\mathcal{O}_S$ be a holomorphy ring of $F/K$. Then $F$ is ...
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0answers
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Question on valuation axioms - Relating to $\mathbb{R}$

In an earlier thread, I asked if there was a standard generalization of the absolute value of $\mathbb{C}$ that could be placed on a field, but might not take values in $R_{\geq 0}$. What somebody ...
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0answers
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Centers of divisorial valuations on toric varieties

Suppose we are given a divisorial valuation $\mathcal{v}$ on a smooth toric variety $X_{\Sigma}$ (for a fan $\Sigma$), i.e. $\mathcal{v}$ is the valuation induced by a torus invariant divisor ...
3
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1answer
32 views

Valuation of a particular element

I am tying to compute the valuation of a particular element of $\mathbb{Q}_p$. I am trying to compute $\operatorname{val}_p(P)$ where $P=\frac{\log(1+p^2)}{\log(1+p)}$ and $\log$ is the $p$-adic ...
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0answers
90 views

A slightly stranger Hensel's Lemma

I'm trying to understand the solution to this problem. It came up when doing some revision. It is essentially to show that the conclusion of Hensel's Lemma holds if we have take a valuation ring $R$ ...
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0answers
46 views

Smooth morphism and completion of DVR

Let $R$ be a Henselian discrete valuation ring with algebraically closed residue field and $\hat{R}$ its $m$-completion, where $m$ is the maximal ideal. Is it true that the induced morphism ...