Euclidean domain

In mathematics, especially in ring theory, a Euclidean domain (also called a Euclidean ring) is a special type of integral domain. It uses a method called a Euclidean function. This method works like Euclidean division with integers.

With this method, we can find the greatest common divisor of any two numbers. This greatest common divisor can always be written as a mix of the two numbers, following Bézout's identity.

Euclidean domains are useful. They help us solve many problems, especially in computer algebra.

We can also think about Euclidean domains compared to another group called principal ideal domains (PIDs). PIDs are similar but sometimes do not have the quick tools to find greatest common divisors.

In every Euclidean domain, each group of numbers (called an ideal) is led by a single number. This means Euclidean domains also follow a rule like the fundamental theorem of arithmetic: they are unique factorization domains. Euclidean domains are part of a bigger family of number systems:

rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields

Definition

In math, a Euclidean domain is a special kind of number system. In this system, you can divide numbers almost like you do with regular whole numbers. It has a rule that helps you break down bigger numbers into smaller parts. This makes it easier to solve problems in these number systems.

Properties

In a Euclidean domain, special rules help us break down numbers and find common factors, just like we do with regular whole numbers. This makes it easier to work with these numbers and learn more about them.

Also, in these domains, some numbers can be turned around to become "invertible." This means they can be used in special ways in calculations. If we can follow steps to find parts of a number, we can learn even more about those numbers.

Norm-Euclidean fields

Algebraic number fields have a special way to measure values, called a norm. This norm helps us understand the basic building blocks of these number fields. When this norm works well, we call the field norm-Euclidean or simply Euclidean.

Some number fields are Euclidean even if their norm doesn’t work perfectly. Others aren’t Euclidean at all. For example, the Gaussian integers — which deal with numbers like -1 — are norm-Euclidean. The norm-Euclidean quadratic fields have been carefully studied, and they include specific values.