Properties of transformation matrices between the normal bases in orders of cubic field

Abstract

ABSTRACT. In this paper, the properties of transformation matrices between the normal bases in orders of algebraic number fields of degree 3 are shown. Let K be a finite extension of the rational number field Q and let the Z K be integral closure of the ring Z in K -i.e. the ring of integral numbers of the field K.Ò Ø ÓÒ 1º Let K be an algebraic number field of degree n over the rational numbers Q. A Z-module B ⊂ K is called an order of the field K if B satisfies the following conditions:(2) B has a basis over Z consisting of n elements, (3) B is ring.Let K be a cyclic algebraic number field of degree n over Q. Such a field has a normal basis over Q, i.e. a basis which consists of all conjugations of one element.Transformation matrices between two normal bases of K over Q are exactly regular rational circulant matrices of degree n.In