Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2004. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 31)

KOSTRA, J. and VAVROS, M., On transformation matrices connected to normal bases in rings

46 J. Kostra, M. Vavros Proposition 1. Let K be a cyclic algebraic number fíeld of degree n over rational numbers. Let A = circ

n(ai,Ű2,...,

a

n) be a circulant matrix and ai, .. •, a

n G Z. By Ai, i — 1,2,

... n we denote the algebraic complement of element a; in the matrix A. Let ai + ö2 H f a

n = ±1 and a, = aj (mod /1) for i, j G {1,2,..., n}, where detA gcd(^i, A

2,...,A

n) ' Then the matrix A transforms a normal basis of an order B of the field A to a normal basis of an order C of the field K, where C C B. In the papers [3, 4] previous matrices are characterized by Theorem 3 [4]. Proposition 2. Let G be a multiplicative semigroup of circulant matrices of degree n, satisfying the assumptions of Proposition 1. Let U be multiplicative group of integral unimodular circulant matrices of degree n. Let II be the semigroup of circulant matrices of type circ

n(a, 6,..., 6), such that a + (n — 1)6= ±1. Then G — H • U. 2. Results First we recall the definition of order of algebraic number field. Definition 1. Let K be an algebraic number field and let the degree of the extension A'/Q be equal to n. A Z-module B C A is called an order of the field K if it satisfies the following conditions: 1. 1 G B, 2. B has a basis over 7L consisting of n elements, 3. B is a ring. Remark 1. Matrices from Proposition 1 transform also normal bases rings which have a basis over Z consist ing of n elements to normal bases of their subrings. Such rings we will call semiorders.