For questions related to valuation functions on a field.

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1answer
24 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
29 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 ...
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1answer
20 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
42 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 ...
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1answer
31 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 ...
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1answer
397 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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1answer
47 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
34 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
41 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}, ...
3
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1answer
135 views

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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2answers
39 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
18 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 ...
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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 ...
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2answers
40 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
28 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
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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2answers
48 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 ...
0
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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 ...
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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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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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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
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 ...
2
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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
88 views

Computing a valuation of a field

Assume $k$ is an algebraically closed field, and $x$ and $y$ are transcendental over $k$. I want to compute the valuation ring of $F$, the field of fractions of the ring $A=k[x,y]/I$, where $I=\langle ...
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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
138 views

How to prove that any infinite algebraic extension of a complete field is never complete?

My first idea is using Baire category theorem since I thought an infinite algebraic extension should be of countable degree. However, this is wrong, according to this post. This approach may still ...
4
votes
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 ...
3
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1answer
34 views

Is $\mathcal O^{\ast}/U^{(n)}\cong (\mathcal O/\mathfrak p)^{\ast}$ as topological groups?

I have the following: $K$ is a field with discrete valuation $v$, $\mathcal O$ its valuation ring and $\mathfrak p$ the maximal ideal and $U^{(n)}=1+\mathfrak p^n$ the $n$-th unit group for $n\geq 1$. ...
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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
votes
1answer
67 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 ...
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1answer
59 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
39 views

Why is the separable closure of a field in it's completion not complete.

I am currently reading "Algebraic Number Theory" by Neukirch and I am a bit confused by what is here. It is on page 143, it states this: $(K,v)$ is a nonarchimedian valued field and ...
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1answer
19 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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0answers
49 views

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
63 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
46 views

How do we extend the valuation on $K[x]$ to a valuation on $K(x)$?

Given that $v$ is a non-archimedean valuation on $K$, we can extend it to $|\cdot|:K[x]\to\mathbb{R}$ by $|a_0 + a_1x + \cdots +a_nx^n|=\max\left\{|a_1|,\ldots,|a_n|\right\}$. My question is how ...
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vote
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
23 views

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 ...