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Questions tagged [valuation-theory]

For questions related to valuation functions on a field.

13
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1answer
385 views

Why is $\mathbb{C}_p$ isomorphic to $\mathbb{C}$?

I know that two closed fields of caracteristic $0$ and uncountable are isomorphic iff they have the same cardinality. But I don't know why $\mathbb{C}_p$ has the same cardinality as $\mathbb{C}$. Can ...
11
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2answers
1k views

Concrete examples of valuation rings of rank two.

Let $A$ be a valuation ring of rank two. Then $A$ gives an example of a commutative ring such that $\mathrm{Spec}(A)$ is a noetherian topological space, but $A$ is non-noetherian. (Indeed, otherwise $...
10
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4answers
644 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 $\...
8
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2answers
1k views

Why is the restricted direct product topology on the idele group stronger than the topology induced by the adele group?

I am confused by the saying that the restricted direct product topology on the idele group is stronger than the topology induced by the adele group. And perhaps this is elaborated by the following ...
8
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2answers
315 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 ...
8
votes
1answer
419 views

Two discrete valuation rings, one contained into another

Let $A$ and $B$ be discrete valuation rings of the same field of fractions. Suppose $A \subset B$. Then $A = B$? I came up with this problem when I was reading van der Waerden's Algebra. The ...
8
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0answers
280 views

An infinite prime can ramify right? (So what is Neukirch talking about?)

I have been under the impression for several years that if $L/K$ is an extension of number fields, then an infinite place of $K$ is said to ramify in $L$ if it comes from a real embedding of $K$ which ...
7
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2answers
377 views

Applications of valuation rings

Some background: I am in the process of writing a research paper for an undergraduate abstract algebra course. I've chosen to write my paper on valuation rings and discrete valuation rings. The goal ...
7
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1answer
393 views

Definition of algebraic variety

In general, the definition of an algebraic variety differs from one reference to other. The definition that I was used to is to consider an algebraic variety as an integral scheme of finite type. ...
7
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1answer
157 views

Is every algebraically closed subfield of $\mathbb C[[X]]$ contained in $\mathbb C$?

Let $F$ be a subring (with same unity) of $\mathbb C[[X]]$ such that $F$ is an algebraically closed field; then is it true that $F \subseteq \mathbb C$ ? Since $F, \mathbb C$ are algebraically ...
7
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1answer
173 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 ...
7
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1answer
220 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 $R_L$ ...
7
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2answers
370 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 ...
7
votes
1answer
215 views

The number of solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}$.

The number of solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}=\{\sum_{i=0}^\infty a_ip^i:0\le a_i \le p-1\}$. I was trying to ...
7
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1answer
624 views

The role of valuation rings in algebraic geometry

I am familiar with basic algebraic geometry in the tradition of Hartshorne's book. Discrete valuation rings appear there in the criteria for separatedness/properness, and are used to define the order ...
7
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1answer
130 views

$(\varphi,\Gamma)$-modules and valuation in Zp

The theory of $(\varphi,\Gamma)$-modules and a reciprocity law due to Cherbonnier and Colmez involve the ring $A_K$, that is described below. The ring $A_K$: Set $E$ to be the set of sequences $(x^{(...
7
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0answers
299 views

Ramification index of infinite primes

I am reading Neukirch's Algebraic Number Theory. On page 184, Chapter 3, Neukirch defines the ramification index of infinite primes as follows: For a finite extension $L/K$ of number fields, and an ...
6
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4answers
1k 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 ...
6
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2answers
829 views

Is every local ring a valuation ring?

Is every local ring a valuation ring? I know the answer is no and the first example comes to my mind was as following (I started with smallest fields, as $\mathbb{Z}_2$ and $\mathbb{Z}_3$ are not ...
6
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1answer
159 views

A certain valuation of $k(X,Y)$ with value group $\mathbb{Z}+\mathbb{Z}\alpha$

Let $k$ be a field, $X$ and $Y$ indeterminates, and suppose that $\alpha$ is a positive irrational number. Then the map $\nu:k[X,Y]\rightarrow \mathbb{R}\cup \{\infty\}$ defined by $\nu\left(\sum c_{n,...
6
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1answer
330 views

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) $v(a+b)\...
6
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1answer
301 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, ...
6
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1answer
427 views

Valuation ring of $k(x, y)$ of dimension $2$

My question is as follows: Given a field $k$, is it always possible to find a valuation ring of $k(x, y)$ of dimension $2$?
6
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1answer
113 views

Geometric interpretation of a result from commutative algebra

I have come across the following result in Hartshorne, Algebraic Geometry, I.6.5 for those who have the book. The result says that if $K$ is a finitely generated extension of some base (...
6
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2answers
935 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 ...
6
votes
1answer
183 views

Lang's “General Integrality Criterion”

Theorem 3.7 in the chapter on ring extension on page 352 of the latest edition of Lang's "Algebra" appears redundant in its phrasing to me. Specifically, if $g_s$ is a polynomial of total degree $<...
6
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0answers
161 views

(Sanity check) the adic spectrum $\operatorname{Spa}(\mathbb{Z}, \mathbb{Z})$

I am following Scholze's and Weinstein's notes on $p$-adic geometry on http://www.math.uni-bonn.de/people/scholze/Berkeley.pdf, example 2.3.6 (page 18 as of this moment bar any updates). $\...
6
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0answers
84 views

If $\xi$ is unramified at the valuation $v$, does the associated homogeneous space $C/K$ have a $K_v$-rational point?

I'm looking at the following Corollary/Remark $(4.4-4.5)$ in Chapter X of Silverman's Arithmetic of Elliptic Curves: Let $\phi:E/K\to E'/K$ be an isogeny defined over $K$, and $S$ a finite set of ...
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0answers
152 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 $R$ ...
6
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0answers
108 views

Isomorphism of algebras $E_v = \prod_{w\mid v} E_w$

I am currently reading the article Bayer–Fluckiger, Eva, B. Nivedita, and Raman Parimala. "Hasse principle for G–quadratic forms." Documenta Mathematica 18 (2013): 383-392. in which there is a ...
5
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2answers
445 views

Does every local domain admit an injective homomorphism into a discrete valuation subring of its fraction field?

Let $R$ be a local ring which is an integral domain with fraction field $K$. Does there exist a discrete valuation subring $D\le K$ containing $R$?
5
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1answer
495 views

How to show the only absolute value on a finite field is the trivial one.

Define the trivial absolute value $|\cdot|$ by $|x| = 1$ if $x \neq 0$ or $|x| = 0$ if $x=0$. The textbook I'm currently reading (Gouvêa - P-adic Numbers An Introduction) asked me to show that for a ...
5
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1answer
292 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 $R/\...
5
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1answer
56 views

A character on $\mathbb A_K$ induces a divisor

Preliminary definitions: Let $K$ be an algebraic function field over $\mathbb F_p$ i.e. $K$ is a finite extension of $\mathbb F_p(t)=K_0$. For any discrete valuation $v$ on $K$, a character on $K_v$ ...
5
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1answer
65 views

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 ...
5
votes
1answer
130 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 primes:...
5
votes
1answer
573 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 fact ...
5
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1answer
754 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 ...
5
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1answer
120 views

Valuations of integer valued polynomials

Consider the ring $R=\text{Int}(\mathbb Z):=\{p(x)\in \mathbb Q[x]\ |\ p(n)\in \mathbb Z, \forall n\in \mathbb Z \}$. Let $K$ denote the fraction field of $R$. Fix an $a\in \mathbb Z$ and let $P$ ...
5
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1answer
252 views

Discrete valuation ring extension such that $A[\pi]$ is not integrally closed

Let $B/A$ be a finite integral extension of discrete valuation rings of characteristic zero. Question. Is there a uniformizer $\pi$ in $B$ such that $A[\pi]$ is a discrete valuation ring? If not, ...
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1answer
95 views

Is the residue field of an algebraically closed valued field algebraically closed?

Is the residue field of an algebraically closed valued field $K$ with valuation ring $A$, algebraically closed? If I take a polynomial $f(x)$ of $k_A$ since $K$ is algebraically closed it has a root ...
5
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1answer
242 views

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 $(O,\mathcal{M})$ on $...
5
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1answer
130 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 ...
5
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1answer
235 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 ?...
5
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0answers
144 views

Is the topology of the p-adic valuation to the unramfied extension discrete?

Consider $\mathbb Q_p^{\text{ur}}$ the maximal unramified extension of the p-adic numbers. Suppose that on $\mathbb Q_p$ we have the usual absolute value that extends $|\frac{a}{b}|_p=\frac{1}{p^{v_p(...
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0answers
69 views

Automorphism of $L|K$ mapping 3 distinct rational points of $S_{L|K}$ to other 3 distinct ones

Let $K$ be a field and consider $L = K(x)$ the field of rational functions. Let $v_{1}, v_{2}, v_{3}$ rational points in the abstract Riemann surface $S_{L|K}$, distinct from each other, and $w_{1}, ...
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2answers
2k views

Roots of unity in a local field

The multiplicative group of a local field $K$ with valuation ring $\mathcal{O}$ and residue class field of $\overline{K}$ of degree $q=p^f$ splits as $K=\langle \pi\rangle\times \mu_{q-1}\times U^{(1)...
4
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2answers
1k 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 $A_i$...
4
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1answer
63 views

Galois Groups in Ramification Theory

I had a slight confusion about Galois groups over a base field which is complete with respect to a discrete valuation. We know that there are irreducible polynomials such as $X^3+X^2+2X-8$ where ...
4
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2answers
237 views

Proof that a DVR is a Euclidean domain

I am independently studying valuation rings and I am currently trying to prove that a DVR is a Euclidean domain using the definition here http://abstract.pugetsound.edu/aata/section-factorization-...