# Questions tagged [valuation-theory]

For questions related to valuation functions on a field.

428 questions
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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 $K_v$ ...
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### 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 ...
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### 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:...
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### 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 ...
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### 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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### 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 $P$ ...
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### 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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### 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 ...
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### 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 $A_i$...
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### 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 ...
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### 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-...