The most important class of simple rings is the category of fields, denoted fields. These are the rings in which every nonzero element is a unit, and, like the quaternions, may be noncommutative. They have also been called division rings, but we retain this terminology only if R is a finite dimensional algebra over its center, in which case we say division algebra. The literature on division algebras is vast, and we refer the reader either to Albert, or to Deuring for an account up to about 1936. Recently, Amitsur solved the question of the existence of division algebras which are not crossed products in the negative, so the reader may refer to this paper for new perspectives.

Steinitz extensions

Steinitz determined that any commutative field F can be obtained as an extension of the prime subfield P by two intermediate extensions: first, a purely transcendental extension T/P (meaning that T is generated by a set {xi}i ∊ I of elements any finite subset of which generates a rational function field over P in those variables); second, F is an algebraic extension of T in the sense that every element y ∊ F satisfies a nonzero polynomial over T.