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

50 J. Kostra, M. Vavros 1 36 ,_i _ . /1111 A ~

circ

5{ 12'36'36'36' and R

2 = (£i, £2, £3, £4, £5), S

2 = (<*i, »2, »3,04, <*s), where (Í*i, t*

2, a

3, a

4, a

5) = (£1,£2,£3, £4, £5) • A, Qi = 9£

2 + 9^3 + 9£

4 + 9S

5 , «2 = 9ei + 9e

3 + 9e

4 + 9e

5 . Then a 1 • a

2 = 8l£i — 81 £4 - 8U5. After transformation by matrix A1 we have 45 9 9 81 81 01-02 = ——01 - 7«2 - -OC3 r«4 T-Q54 4 4 4 4 From this it follows that S

2 is not a ring. And now let Ri = (£1, £2, £3, £4, £5) and Si — (ßi, ß

2, /?

4, ß

5), where ßi = 6£

i , ß'2 - 6£

2 , /?3 = 6^3 , ß

4 — 6^4 , ߧ = 6e

5

. <7l,72,73,74,75> = (ßl,/3

2, #3,/?4, A>) • A We have 7i7j = 36 • (&i/3i + 62/^2 + 1- Hßs)- From the expression of A1 it follows that 7t7j = C171 + C272 H h C575 with integral rational coefficients c*. So is a semi order. References [1] BOKEVICH, Z. I., SHAFAREVICH, í. R,., Number theory, Nauka, Moscow, 1985. 3rd ed. (in Russian). [2] DAVIS, P. J., Circulant matrices, A. Wiley-Interscience Publisher, John Wiley and Sons, New York-Chichester-Brisbane-Toronto, 1979.