What is euclidean domain?

In mathematics, an Euclidean domain is a type of ring that satisfies certain properties related to division with remainder. Specifically, an Euclidean domain is an integral domain in which a division algorithm exists, meaning that given any two elements a and b in the domain with b nonzero, there exist unique elements q and r, called the quotient and remainder, respectively, such that a = bq + r and either r = 0 or the degree of r (i.e., its "size") is strictly less than the degree of b.

One of the key implications of a ring being an Euclidean domain is that it guarantees the existence of greatest common divisors for any pair of elements. This property is particularly useful in areas such as number theory and algebraic geometry.

Examples of Euclidean domains include the ring of integers, polynomial rings over a field, and Gaussian integers. Euclidean domains are also principal ideal domains, meaning that every ideal in the ring can be generated by a single element. This property further enhances their usefulness in various mathematical contexts.

Overall, Euclidean domains play a fundamental role in algebraic structures and provide important tools for solving problems in algebra and number theory.