Polynomial ring over a field

Definition

The polynomial ring over a field is defined as the polynomial ring whose base ring is a field.

Facts

Filtrative Euclidean domain

The degree function on the polynomial ring over a field, that associates to every polynomial its degree, defines a Euclidean norm on the polynomial ring, turning it into a Euclidean domain. In fact, it is both a characteristic Euclidean domain and a filtrative Euclidean domain under this norm.

As a consequence of being a Euclidean domain, the polynomial ring over a field is thus also a principal ideal domain, a Noetherian ring, a unique factorization domain and a normal domain.