For questions related to valuation functions on a field.

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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 $u\in A$? ...
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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 ...
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1answer
42 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 ...
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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 ...
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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 $0$ ...
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1answer
58 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
67 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 $x\in A$ ...
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2answers
131 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 ...
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1answer
106 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 $R_L$ ...
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0answers
77 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 ...
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1answer
105 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 ...
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1answer
184 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 ...
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1answer
56 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 ...
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1answer
254 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 ...
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2answers
138 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 ...
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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 ...
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3answers
260 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 ...
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126 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, ...
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2answers
157 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 ...
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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\{ ...
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1answer
157 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) ...
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45 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 $\mathcal{O}_L$ ...
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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 ...
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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 ...
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2answers
87 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 $m_A$. ...
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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 ...
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1answer
109 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 ...
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1answer
124 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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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?
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2answers
255 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 ...
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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 ...
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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 $D$ ...
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1answer
102 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 ...
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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 $\mathbb Q$ ?
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1answer
537 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 ...
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1answer
98 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
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220 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 ...
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2answers
169 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 ...
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1answer
135 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 ...
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1answer
267 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 ...
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1answer
148 views

What is a valuation associated to an ordering on a field?

If $(K,\leq)$ is a totally ordered field with $P\!=\!\{\alpha\!\in\!K;\, 0\!\leq\!\alpha\}$, how is the valuation associated to $P$ defined? I was searching through Prestel & Delzell's Positive ...
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1answer
148 views

A property of non-Archimedean metrics

I have recently been reading about non-Archimedean metrics on fields (in Koblitz: $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions), and came across the exercise: Prove that a norm $\|.\|$ on ...
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1answer
285 views

Existence of valuation rings in a finite extension of the field of fractions of a weakly Artinian domain without Axiom of Choice

Can we prove the following theorem without Axiom of Choice? This is a generalization of this problem. Theorem Let $A$ be a weakly Artinian domain. Let $K$ be the field of fractions of $A$. Let $L$ ...
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424 views

Why does the equation $x^2-82y^2=\pm2$ have solutions in every $\mathbb{Z}_p$ but not in $\mathbb{Z}$?

I have been working on an exercise in H. P. F. Swinnerton-Dyer's book, A Brief Guide to Algebraic Number Theory. The question is like this: Show that $x^2-82y^2=\pm2$ has solutions in every ...
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1answer
96 views

Ramification of local field

Let $K$ be finite field, $E=K((x))$ and $F=K((x^n))$, that is $F$ is field of formal laurent series in $x^n$. I know that $E\cong F$ (because you can consider $x\rightarrow x^n$ ) I want to prove if ...