Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)
NYUL, G., Power integral bases in mixed biquadratic number fields
Acta Acad. Paed.. Agriensis, Sectio Mathematicae 28 (2001) 79-113 POWER INTEGRAL BASES IN MIXED BIQUADRATIC NUMBER FIELDS Gábor Nyul (Debrecen, Hungary) Abstract. We give a complete characterization of power integral bases in quartic number fields of type A = Q( \/?77, \/n) where m,7l are distinct square-free integers with opposite sign. We provide a list of all fields of this type up to discriminant 1Ü 4 in increasing order of discriminants containing field indices, minimal indices and all elements of minimal index. AMS Classification Number: 11D57, 11Y50 Keywords: quartic number fields, power integral bases 1. Introduction Let K be an algebraic number field of degree n. The index of a primitive element a E TJK is defined by 1(a) = (Z+ : Z[a] +). The existence of power integral bases {1, a, ... , cr"1} is a classical problem of algebraic number theory. The element a generates a power integral basis if and only if 1(a) = 1 (for related results cf. [1]). If the number field K admits power integral bases, it is called vionogeneous. We recall that the ininimal index of a number field K is the minimum of the indices of till primitive integers in the field. The field index is the greatest common divisor of the indices of all primitive integers of the field. Let m, n be distinct square-free integers. Biquadratic fields of type K — Q(^/rn, ^/n) were considered by several authors. K. S. Williams [6] described an integral basis of K. T. Nakahara [5] proved that infinitely many fields of this type have power integral bases, on the other hand for any given N there are infinitely many fields of this type with field index 1 but minimal index > N , consequently without power integral basis. M. N. Gras and F. Tanoe [4] gave necessary and sufficient conditions for biquadratic fields to have power integral basis. In fact they characterized all mixed biquadratic fields having power integral basis and established further necessary conditions for totally real biquadratic fields to have power integral basis. Using the integral bases I. Gaál, A. Pethö and M. Pohst [3] formulated the corresponding
