Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-12T21:13:51.496Z Has data issue: false hasContentIssue false
  • English
  • Français

Integers of Biquadratic Fields

Published online by Cambridge University Press:  20 November 2018

Kenneth S. Williams
Kenneth S. Williams*
Affiliation:
Carleton University, Ottawa, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 4 , December 1970 , pp. 519 - 526
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Hancock, H., Foundations of the theory of algebraic numbers, Dover, N.Y., Vol. 2 (1964), 392-396.Google Scholar
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. Pollard, H., The theory of algebraic numbers, Carus Math. Monograph, No. 1, M.A.A. Publ. (1961), 61-63.Google Scholar