For questions related to valuation functions on a field.

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7
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1answer
121 views

A valuation-like function $w: \mathbb{N}^{+} \rightarrow \mathbb{N}$ is a $p$-adic valuation?

This question is a variant of problem 4, pg. 21, from Birkhoff and Maclane, A Survey of Modern Algebra. Given a function $w: \mathbb{N}^+ \rightarrow \mathbb{N}$ that behaves like a valuation ...
2
votes
2answers
115 views

Proving that a discrete valuation-like function $w: \mathbb{Z}\backslash\{0\} \rightarrow \mathbb{N}$ is a $p$-adic valuation

This problem is from Birkhoff and Maclane, A Survey of Modern Algebra, pg 21, problem 4*. Given a function $w: \mathbb{Z}\backslash\{0\} \rightarrow \mathbb{N}$ that behaves like a discrete ...
1
vote
1answer
30 views

A certain ideal of a valuation ring

A question from Frohlich and Taylor's book 'Algebraic Number Theory', page 60. Let $\mathfrak o$ be a Dedekind domain with quotient field $K$ and let $v$ be a discrete valuation on $K$. Let ...
0
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1answer
33 views

Two discrete valuations on a Dedekind domain

A question from Frohlich and Taylor's book 'Algebraic Number Theory', page 60. Let $\mathfrak o$ be a Dedekind domain with quotient field $K$ and let $v$ be a discrete valuation on $K$. Let ...
2
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0answers
70 views

How to prove that a DVR is not complete

My question is inspired by a comment in this topic. How to prove that $R=\mathbb C[x]_{(x)}$ is not complete in the topology of its maximal ideal? One knows that $R$ is a DVR, and its field of ...
0
votes
0answers
16 views

Division in a complete subring of a local field

Suppose $A$ is a complete subring of a local field such that a prime element $\pi$ belongs to $A$. Is it true that if $\beta=\pi^k u$ (with $k\ge 0$ and $v(u)=0$) and $\beta\in A$ then also $u\in A$? ...
1
vote
2answers
41 views

Show $m_v$ is maximal in $R_v$, valuation ring ($v:K\rightarrow G\cup\{\infty\}$ a valuation on $K$)

Let $K$ be a field, $G$ a totally ordered group and $v:K \rightarrow G\cup\{\infty\}$ be a valuation on $K$. I am trying to show that $m_v=\{k \in K: v(k)>0\}$ is a (the?) maximal ideal in ...
1
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1answer
43 views

Simple property of a valuation on a field

Let $F$ be a field and $v:F\rightarrow G\cup\{\infty\}$ be a valuation on $F$ so $G$ is a totally ordered abelian group with $\infty$ having the properties $\infty+\infty=g+\infty=\infty+g=\infty$ and ...
0
votes
1answer
38 views

How to show that $R_{\mathfrak{m}}$ is $R$?

Let $R$ be a discrete valuation ring and $\mathfrak{m}$ its unique non-zero maximal ideal. How to show that $R_{\mathfrak{m}}$ is $R$ using definition of a discrete valuation ring? I know that ...
3
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0answers
78 views

Question about localizations of discrete valuation rings.

Let $R$ be a discrete valuation ring. Then $R$ has only two prime ideals: $0$ and the maximal ideal $\mathfrak{m}$. It is said in Hartshorne, page 74, Example 2.3.2 that the localization of $R$ at $0$ ...
3
votes
1answer
61 views

Regarding valuation on function field

Notations: Let $k$ be a field of characteristic $\neq 2$, $k(X)$ be a function field. Suppose $p(X)\in k[X]$ is an irreducible polynomial and $\alpha$ be a root of $p(X)$. Let ...
0
votes
2answers
78 views

Primes corresponding to embeddings of a number field

Let $k$ be a number field. Define a prime of $k$ to be an equivalence class of absolute values on $k$. If $\sigma:k\hookrightarrow \mathbb{C}$ is an embedding of $k$ into the complex numbers then we ...
2
votes
1answer
53 views

Integral elements of a non-archimedean completion of a number field

Let $k$ be a number field, $v$ a discrete non-archimedean valuation on $k$. Let $k_v$ be the completion of $k$ wrt $v$ and $\mathcal O_v$ its valuation ring. My question is: If $x\in k_v$ then is ...
0
votes
1answer
27 views

Module Notation $Dx$

Let $K$ be a field, and let $D$ be a subring of $K$ with identity. Let $K^*$ be the multiplicative group of nonzero elements of $K$. The group $U$ of units of $D$ is a subgroup of $K^*$. We take ...
1
vote
1answer
27 views

Quotients of a valuation ring in the completion of a number field

Let $k$ be a number field, $v$ a discrete (non-archimedean) valuation on $k$. Let $k_v$ be the completion of $k$ with respect to $v$. Also let $\mathcal{O}_v$ be the valuation ring of $k_v$ and ...
0
votes
1answer
69 views

discrete valuation ring

I am struggling to understand the proof of the following proposition Let $A=\{x\in K|v(x)\ge 0\}$ for a field $K$ be a discrete valuation ring. Let $t\in A$ s.t. $v(t)=1$. Then any element $x\in A$ ...
3
votes
2answers
135 views

Learn about valuations, valuation rings, value group

I am reading a paper for a summer research project (Example of an interpolation domain ). I am unfamiliar with some of the terms used here and I have tried searching on google for definitions but I am ...
6
votes
1answer
108 views

Valuations on a field and ramification

For $K\subseteq L$, where $L$ is a finite number field extension of $K$, we consider $p\subset R_K$ and $p'\subset R_L$ where $p'$ lies over $p$, where $R_K$ is the ring of integers of $K$ and $R_L$ ...
0
votes
0answers
81 views

Extensions of valuations

I'm trying to understand how a valuation $v$ of a field $K$ extends to an algebraic extension $L$. In chapter 8 of his book ANT, Neukirch first chooses a $K$-embedding $\tau : L \longrightarrow ...
5
votes
1answer
106 views

Valuation ring in an algebraically closed field

Let $k$ be an algebraically closed field and $(R, \mathfrak{m})$ a valuation ring in $k$ i.e. the field of fractions of $R$ is $k$. Then Mumford (Red Book, page 127) claimed that the residue field ...
1
vote
1answer
192 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 $|\cdot|_{1},\cdots, |\cdot|_n$ be pairwise non-equivalent nontrivial ...
2
votes
1answer
57 views

extension of a valuation of $K$ to $K(X)$

Let $A$ be a Krull ring and $p$ a prime ideal of height 1. Then $A_p$ is a DVR with corresponding valuation $v$ on the field of fractions $K$ of $A$. Question: Can we extend this valuation to an ...
1
vote
1answer
260 views

How do we find prime ideals of a ring of integers of a number fileld?

For example for $F=Q(\sqrt{-5})$. the ring of integers of $F =Z[\sqrt{-5}]$.(since $-5\equiv3 \pmod 4$) but how can we determine prime ideals of this? and another problem is the corresponding ...
2
votes
2answers
142 views

Local fields and infinite extensions, basic questions

Notation throughout: Let $K$ be a discrete valuation field and $L/K$ an infinite (not necessarily Galois) extension of $K$. 1) How can/does one define a ramification index $e(L/K)$ for $L/K$? It ...
3
votes
1answer
121 views

Discrete Valuation Ring and Subring of the Fractions Field

Let $R$ be a Discrete Valuation Ring, and $K$ its fractions field. Now if $B\subseteq K$ is a subring with $R\subseteq B$ then we have $$B=R \text{ or } B=K.$$ Now this seems to be a very basic ...
2
votes
3answers
268 views

Examples of Non-Noetherian Valuation Rings

For valuation rings I know examples which are Noetherian. I know there are good standard non Noetherian Valuation Rings. Can anybody please give some examples of rings of this kind? I am very ...
4
votes
0answers
127 views

Intuition behind “Non-Archimedean” — two senses of “non-archimedean”.

There appear to be two senses of the qualifier "Archimedean" for fields. One is for ordered fields, and one is for "valued fields" (fields with an absolute value function defined). In the first case, ...
2
votes
2answers
160 views

The composite of all unramified extensions inside an algebraic closure

I'm reading Ch.II, $\S$ 7 of Neukirch's Algebraic Number Theory and I'd be really grateful if someone could help me understand the following: Let $K$ be a complete valued field wrt a non-archimedean ...
2
votes
2answers
63 views

Valuations, Isomorphism, Local ring

Let $x \in \mathbb{Q}, \, x \neq 0$, such that $x=p^r \frac{a}{b}, \quad a,b, r \in \mathbb{Z}, \quad p \nmid a, \quad p\nmid b$. Let $v_p(x):=r$ and $v_p(0):= \infty$. Also, $$\mathcal O_p= \left\{ ...
0
votes
1answer
161 views

Discrete Valuation Rings problem 2

An order function on a field $K$ is a function $\phi:K\to \mathbb{Z} \cup {\{\infty}\}$ satisfying: i) $\phi(a) = \infty$ if and only if $a=0$. ii) $\phi(ab) = \phi(a) + \phi(b)$. iii) ...
1
vote
0answers
46 views

A trace formula for an extension of complete valued fields

I'm trying to understand the following fact. Proposition. Let $L/K$ be a finite separable extension of complete valued fields, with respective valuations $w$ and $v$, valuation rings $\mathcal{O}_L$ ...
2
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0answers
48 views

A question on a sum of valuations

Let $A$ be a discrete valuation ring of characteristic zero. Let $v$ be the valuation on $A$. Let $I$ be a finite index set and $d_i$ a positive integer for all $i$ in $I$ and define $$ d:= \sum_{i ...
4
votes
1answer
77 views

About Henselization

I have some question about Henselization of valued field. If $(K_{1}, \nu_{1})$ is a Henselization of valued field $(K, \nu)$. Which one is true. $K_{1}/K$ is an algebraic extension. $K_{1}/K$ is a ...
2
votes
2answers
89 views

Valuation rings and domination

I was wondering if the following holds: if $A$ is a valuation ring with maximal ideal $m_A$ and $B$ is a ring extension of $A$ contained in $Frac(A)$ then the only maximal ideal of $B$ is still $m_A$. ...
2
votes
1answer
72 views

Properties of valuation map

It seems not so hard to prove but how can we prove by induction. Let $K$ be a field and $\nu$ be a valuation map. If $a_{1} + a_{2} + ... + a_{n} = 0$ then prove that $\nu(a_{i}) = \nu(a_{j})$ for ...
1
vote
1answer
113 views

A question in valuation theory

Let $R$ be an integral domain with quotient field $K$. Let $P$ be a prime ideal of $R$, and $L$ be a transcendental extension of $K$. I think there is more than one valuation domain $O'$ with quotient ...
2
votes
1answer
126 views

Construction of a Valuation Ring via Zorn's Lemma, except not

In Atiyah & MacDonald they provide an abstract way to construct valuation rings, I'm curious how easy it is to work this out in general. As a refresher let $K$ be a field, $L$ an algebraically ...
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2answers
85 views

Function field has infinitely many valuations.

Suppose we have a field $K$ which is a finite algebraic extension of the field $\mathbb{C}(X)$. Can you give me an argument that $K$ admits infinitly many discrete valuations?
7
votes
2answers
258 views

Non-trivial valuation of $\mathbb R$

In a valued fields book on page $82$ there is a question: "show that every non-trivial valuation of $\mathbb R$ has divisible value group and algebraically closed residue class field." How do I ...
5
votes
1answer
148 views

Ring of integers in a field of fractions

Let $R$ be ring with complete non archimedian absolute value. Let $Q$ be the associated field of fractions with the extended absolute value. Does the ring $O_Q = \{x\in Q | |x|\leq 1\}$ is complete ...
3
votes
1answer
77 views

Real valuations on Dedekind domains

Let $D$ be a Dedekind domain. Let $v:D \to \mathbb{R}$ a valuation. We know that for every prime ideal $\mathfrak p$ of $D$ the localization $D_{\mathfrak p}$ is DVR. Does every valuation on $D$ ...
4
votes
1answer
105 views

Henselization and immediate extension

I am reading about the henselization and immediate extension of valuation. I am getting confusion about some basic terminology. I have few question. \ $1)$ Is every hensilization extension of ...
2
votes
1answer
99 views

Valuation over the algebraically closed field of rational number

How do we define the valuation over the algebraically closed field of rational numbers say $\bar{\mathbb Q}$ as an extension of the valuation of $\mathbb Q$ ?
5
votes
1answer
541 views

Extension of valuation

We define a valuation on the field of rational number $\mathbb Q$ as follows. For example if we choose a prime number $2$ then for $x \neq 0\in \mathbb Q$, $v(x) = v(2^{n}a/b)= n$ where $n$ is an ...
1
vote
1answer
101 views

Valuation rings of complete non-archimedean fields which are not local

I would like to know how are the valuation rings of complete non-archimedean fields which are not local. Take, for example, $\mathbb{C}_p$, and its valuation ring, ...
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0answers
69 views

Henselization and valued field

What is the importance of henselization in valuation theory, when the rank of valuation is bigger than one? Thanks
3
votes
2answers
225 views

What is a “normalized valuation” corresponding to a valuation ring?

I encountered the phrase "normalized valuation" similar to the following: Let $A_i$ be the valuation ring $k[x_1,...,x_n]_{\langle x_i\rangle}$ and $v_i$ be the normalized valuation defined by ...
7
votes
2answers
172 views

How many absolute values are there?

My question is the following: Are there algebraic norms on the fields $\mathbb{R}, \mathbf{Q_p}$ ''other'' than the absolute value, respectively $|\cdot|_p$? Now phrasing more precisely: If generally ...
1
vote
1answer
136 views

Formal power series ring over a valuation ring of dimension $\geq 2$ is not integrally closed.

I recently tried exercise 10.4 in Matsumura's Commutative Ring Theory, but got stuck. The question is: If $R$ is a valuation ring of Krull dimension $\geq 2$, then the formal power series ring ...
3
votes
1answer
271 views

Non-Archimedean Fields

Are there non-Archimedean fields without associated valuation or being a non-archimedan field implies it is a valuation field? I understand that a non-Archimedean field is a field which does not ...