Integers of Biquadratic Fields

Williams, Kenneth S.

Canadian mathematical bulletin, Tome 13 (1970) no. 4, pp. 519-526

Résumé

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 x 4 —2(m + n)x 2 + (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 a 1, a 2, a 3 ∊ 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).

Détail

DOI : 10.4153/CMB-1970-094-8

Bibliographie

[1] 1. Hancock, H., Foundations of the theory of algebraic numbers, Dover, N.Y., Vol. 2 (1964), 392-396. Google Scholar

[2] 2. Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, Oxford Univ. Press, London, 4th ed. (1960), 230-231. Google Scholar

[3] 3. Pollard, H., The theory of algebraic numbers, Carus Math. Monograph, No. 1, M.A.A. Publ. (1961), 61-63. Google Scholar