Valuation (algebra)

In algebra, especially in areas like algebraic geometry or algebraic number theory, a valuation is a special function. It helps us understand the size of numbers or math objects in a certain way.

This idea of valuation comes from many parts of math. It is like looking at how many times a number can be divided by another number. In these cases, the valuation gives us whole number answers. 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 better.

Definition

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

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

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.

Basic properties

Two valuations are 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 simple example is the p-adic valuation. It relates to a prime number p. It works with rational numbers and shows how divisible a number is by powers of p.

This idea helps create p-adic numbers. These numbers are important in number theory.

Order of vanishing

In geometry, we study functions on shapes called Riemann surfaces. By looking at how these functions act near certain points, we learn more about their properties. This helps us understand the shape of these geometric objects.

π-adic valuation

We can extend the p-adic valuation to more general number systems. This means breaking down numbers into smaller parts and studying how they relate.

P-adic valuation on a Dedekind domain

We can apply these ideas 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. In these spaces, we study how different parts relate to each other using something called a valuation.

We can talk about sets of points that can "absorb" other sets. This means they can contain them when multiplied by certain values. Some of these sets are special. We call them "radial" or "circled." These sets have unique properties when we combine them or apply math rules.

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