Integers of Biquadratic Fields

Abstract

Let Q denote the field of rational numbers. If m, n are distinct squarefree integers the field formed by adjoining √m and √n to Q is denoted by Q(√m, √n). Since Q(√m, √n) = Q(√m, √n) and √m + √n has for its unique minimal polynomial x4 —2(m + n)x2 + (m - n)2, Q(√m, √n) is a biquadratic field over Q. The elements of Q(√m, √n) are of the form a0 + a1√m + a2√n + a3√mn, where a1, a2, a3 ∊ Q. Any element of Q(√m, √n) which satisfies a monic equation of degree ≥ 1 with rational integral coefficients is called an integer of Q(√m, √n). The set of all these integers is an integral domain. In this paper we determine the explicit form of the integers of Q(√m, √n) (Theorem 1), an integral basis for Q(√m, √n) (Theorem 2), and the discriminant of Q(√m, √n) (Theorem 3).