Euclidean domain

In mathematics, more especially in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can use a special method called a Euclidean function. This method lets us do something similar to Euclidean division of integers. With this tool, we can find the greatest common divisor of any two numbers in the domain. This greatest common divisor can always be written as a mix of the two numbers, following something called Bézout's identity.

Euclidean domains are very useful because they let us use a special algorithm, just like the one Euclid used for integers, to solve many problems. This is important in areas like computer algebra, where we work with numbers and formulas on computers.

We can also compare Euclidean domains to another group of number systems called principal ideal domains (PIDs). While PIDs share many properties with Euclidean domains, they do not always have the tools needed to quickly find greatest common divisors. Knowing whether a number system is a Euclidean domain helps us understand its structure and solve problems more easily.

In every Euclidean domain, each special group of numbers (called an ideal) is led by a single number. This means Euclidean domains also follow a rule similar to the fundamental theorem of arithmetic: they are also unique factorization domains. Euclidean domains fit into a bigger family of number systems, as shown in this chain:

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 where you can divide numbers almost like you do with regular whole numbers. It has a special rule that lets you break down bigger numbers into smaller parts, which helps in finding the biggest number that can divide two others. This makes solving problems in these number systems easier and more organized.

Properties

In a Euclidean domain, special rules help us break down numbers and find common factors, much like how we do with regular whole numbers. This makes it easier to work with these numbers and understand their properties.

Also, in such domains, certain numbers can be turned around to become "invertible," meaning they can be used in special ways in calculations. If we can follow steps to find parts of a number, we can also find even more details about those numbers.

Norm-Euclidean fields

Algebraic number fields come with 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 enough, 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.