Valuation (algebra)

In algebra, especially in areas like algebraic geometry or algebraic number theory, a valuation is a special function that helps us understand the size or how many times elements of a field appear. Think of it like a tool that measures how "big" or "small" numbers or mathematical objects are in a certain way.

This idea of valuation comes from many different parts of math. It is like looking at how many times a number can be divided by another number, or how close two shapes are to touching each other. In these cases, the valuation gives us whole number answers, and we call this a discrete valuation.

When we have a field with a valuation, we call it a valued field. This concept helps mathematicians study numbers and shapes in deeper and more useful ways.

Definition

A discrete valuation is a special way to measure elements in a field, which is a type of mathematical structure. It helps us understand properties like divisibility or how elements come together.

Think of it like a ruler that tells us how "big" or "small" an element is, but instead of length, it tells us about mathematical properties. This idea comes from many areas of math, like how numbers can be divided by primes or how shapes touch each other.

In simpler terms, a discrete valuation follows a few important rules that help us study and compare elements in a field. It links together ideas from number theory, geometry, and algebra in a neat package.

Basic properties

Two valuations are like different ways to measure the "size" of numbers in a field. They are considered the same, or equivalent, if they give the same results after a special kind of matching. This matching is called an order-preserving group isomorphism.

When we extend a field, we can also extend its valuation. This means we find a new way to measure "size" in the larger field that agrees with the original way on the smaller field. These extensions help us understand how the field grows and changes.

A complete valued field is one where this measuring works perfectly, without any gaps. If a field isn’t complete, we can build a bigger field that fills in those gaps.

Examples

See also: Discrete valuation ring § Examples

p-adic valuation

A basic example is the p-adic valuation, which relates to a prime number p. It works with rational numbers and helps us understand how divisible a number is by powers of p.

This idea leads to the creation of p-adic numbers, which are an important part of number theory.

Order of vanishing

In geometry, we can study functions on shapes called Riemann surfaces. By looking at how these functions behave near certain points, we can understand their properties better. This helps in analyzing the structure of these geometric objects.

π-adic valuation

We can extend the idea of p-adic valuation to more general number systems. This involves breaking down numbers into their basic building blocks and studying their relationships.

P-adic valuation on a Dedekind domain

Even more generally, these ideas can be applied to special types of number rings called Dedekind domains. This helps us understand the structure of algebraic number fields and their completions.

Vector spaces over valuation fields

Imagine we have a special kind of math space called a vector space, where we study how different parts of the space relate to each other using something called a valuation. In these spaces, we can talk about sets of points that "absorb" other sets, meaning they can contain them when multiplied by certain values. Some of these sets are special — we call them "radial" or "circled" — and they have unique properties when we combine them or apply certain math rules.

When we have two of these spaces and a rule that connects them, we can see how these special sets behave under that rule. This helps us understand more about the structure and relationships within these mathematical spaces.