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

Acta Academiae Paedagogicae Agriensis, Sectio Mathematical 31 (2004) 4-5 51 ON TRANSFORMATION MATRICES CONNECTED TO NORMAL BASES IN RINGS J. Kostra (Zilina, Slovakia), M. Vavros (Ostrava, Czech Republic) Abstract. In the paper [6, Problem 7] there is presented an open problem to characterize all circulant matrices which transform any normal basis of any order of cyclic algebraic number field K to a normal basis of its suborder in K. A conjecture is that if a circulant matrix A= circ„(ai,a2,...,<Jn), ^ a,=±l, transforms some normal basis of ring to normal basis of its subring 1 = 1 then it. transforms any normal basis of ring to normal basis of its subring. In this paper it is shown that if Y^ then the related conjecture is false. 1=1 AMS Classification Number: 11R16, 11C20 1. Introduction Let K be a tamely ramified cyclic algebraic number field of degree n over the rational numbers Q. It seems that A C Q(Cm)> where (

m is a m-th primitive root of unity and m is square free. Such a field has a normal basis over the rationals Q, i.e. a basis consisting of all conjugations of one element. Transformation matrices between two normal bases of A over Q are exactly regular rational circulant matrices of degree n. In the paper [6, Problem 7] there is presented an open problem to characterize all circulant matrices which transform any normal basis of any order of cyclic algebraic number field A to a normal basis of its suborder in A. A conjecture is n that, if a circulant matrix A - circ

n (01,02,... ,o

n), o, — ±1, transforms some j=i normal basis of ring to a normal basis of its subring, then if transforms any normal basis of ring to a normal basis of its subring. In the paper it, is shown that if n i

2 then the related conjecture is false. 1 = 1 In the paper [5], the special class of circulant matrices with integral rational elements is characterized by the following proposition. This research was supported by VEGA '2/4138/24