Euclidean Quadratic Fields

Abstract

algebraic field Q(a) is the smallest subfield of the complex numbers which contains a. The algebraic integers I(a) are those elements of Q(a) which are roots of monic polynomials with (ordinary) integer coefficients. Computation in I(a) is in general unlike computation in the integers Z, since usually there is no analogue of the uniqueness of prime factorization. Historically the (false) assumption that prime factorization is unique in every algebraic field proved to be a stumbling block for various distinguished mathematicians, among them Gabriel Lame. (A nice discussion is given by Edwards [6, Chap. 4].) The problem of determining the algebraic fields which do have unique factorization is still not completely solved. However, in certain fields, known as Euclidean fields, it is possible to define an analogue of Euclid's algorithm, and in such cases this guarantees unique factorization. The algebraic fields of degree 2 which have this property are called Euclidean quadratic fields. Work of Davenport and others, culminating in 1952, showed that there are just 21 of them. The well-known book by Hardy and Wright [8] is a standard reference on Euclidean quadratic fields. In 14 of the 21 cases they present proofs that Q(d ) is Euclidean. The reader naturally wonders whether the proofs in the 7 remaining Euclidean cases are difficult. Hardy and Wright also prove that there are no other Euclidean cases with d 0. Indeed, Q(a) = Q(Vd ) in this case. We call this the quadratic field with discriminant d. If the discriminant is negative, the field is complex, and otherwise real. It is convenient to refer to it as Q(Vd ), and to refer to its set of algebraic integers as iG/d ).