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
80 Gábor Nyúl index forms and gave an algorithm for determining all generators of power integral bases in the totally real case by solving systems of simultaneous Pellian equations. To complete the above theory of power integral bases in biquadratic fields our purpose is to describe all generators of power integral bases in mixed biquadratic number fields. The most interesting point is that it turns out, that surprisingly the coordinate vectors (with respect to the integral basis of [6]) of the generators of power integral bases in mixed biquadratic number fields are contained in a finite set of constant vectors for all these fields. We also provide a table of mixed biquadratic fields in increasing order of discriminants up to 1Ü 4 displaying the field index, minimal index and all elements of minimal index. 2, Index form equation in mixed biquadratic fields To fix our notation we shortly recall the integral bases and corresponding index forms of biquadratic number fields with mixed signature. Let m, n be distinct square-free rational integers (not equal to 1), let / = (m, n) > 0 and let nil . n\ be defined by m = /mi, n — ln\. By K. S. Williams' result [6] all mixed biquadratic number fields can be given in the form Qfy^m, \/n) so that the parameters belong to one of the following cases: Case 1: Case 2: Case 3/A Case 3/B Case 4/A Case 4/B Case 5/A Case 5/B and the integral bases are given by 1 + \/rn 1 + x/n 1 + / m + y/n + /nnm m >0, n < o, in = 1 (mod 4), n = 1 (mod 4), mi = 1 (mod 4), n i = 1 (mod 4). m > o, n < o, in = 1 (mod 4), n = 1 (mod 4), mi =3 (mod 4), n j =3 (mod 4). m >0, n < o, in =1 (mod 4), n =2 (mod 4). m <0, n > o, m =1 (mod 4), n =2 (mod 4). m > Ü, n < o, m =2 (mod 4), n =3 (mod 4). rn < 0, n > o, m =2 (mod 4), n =3 (mod 4). m > 0, n < o, m =3 (mod 4), n =3 (mod 4). m < 0, n < o, m =3 (mod 4), n =3 (mod 4). Case 1: <1 Case 2: 1 2 ' 2 ' 4 1 + x/m. 1 + y/n 1 - \/rn + /n + \/ mi ni Cases 3/A and 3/B: Cases 4/A and 4/B: Cases 5/A and 5/B: 2 1 + n + /mi??; 1, x/m, y/n, m -f- ^/rriirii ~2 f— /m. + y/ri 1 + l.Vm, 2 ' 2
