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

Power integral bases in mixed biquadratic number fields 81 The integral basis enables one to construct the corresponding index forms (cf. e.g. [2]): Case 1: + - J®4j + y) ~-y£

4J ("i(*3 + y) - m

x{x

2 + y) JCase 2: / nl 2 A (u , ,2 ml 2\ ( / , ;

C4 / - y) - y x4J + y) - J + y) - - y) J • Cases 3/A and 3/B: (/^2 - " 1^4) (^(

;t

;3 + y)

2 - "y

x4j («1(2^3 + £4)" - miij). Cases 4/A and 4/B: + - FA! - f M - T^ + *

4)

í)Cases 5/A and 5/B: (/(2.1-2 + Z3)

2 - iHx'l) (Ix

3 - mizj) ( y

;

c3 - mi(®2 + y )

2) • As it is well-known (see e.g. [1]) K = Q(\/m, ^/n) admits power integral bases if and only if the index form equation (1) I(x2, £3, £4) = ±1 (in .r

2, .T

3, x

4 E Z) is solvable, where I(x

2, £3, £4) is the index form given above. Moreover, all generators of power integral bases are of the form a = X\ + X2W0 + .C3CJ3 + X4ŰJ4 where {1, <^2,^3,^4} is the integral basis of K\ (x

2, £3, £4) is a solution of the index form equation (1) and x\ E Z is arbitrary. Our main theorem characterizes the cases when K has power integral bases and describes all generators of power integral bases. M. N. Gras and F. Tanoe [4] has already described the monogeneous mixed biquadratic fields. Our main point is to show that the solutions of the index form equations in monogeneous mixed biquadratic fields belong to a finite set of constant vectors. Especially, the coordinates of the generators of power integral bases are explicitely given and do not depend on the parameters m,n,l.