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

On transformation matrices connected to normal bases in rings 17 Definition 2. Let K be an algebraic number field and let the degree of the extension K/Q be equal to n. A Z-module B C A is called a semiorder of the field K if it satisfies the following conditions: 1. B has a basis over Z consisting of n elements, 2. B is a ring. In the following it will be shown that the condition a + {n - 1)6= ±1. from Proposition 1 for matrix circ

n(«, 6,..., b) is necessary. Example 1. Let (V be a 7-th primitive root of unity and let (e 1,^2, £3) '>e a normal integral basis of the field K = Q

+(CT) over Q, where it - CR + <7, £2 = C

t + C7. -3 = Cr + C

tLet A = circ

3(0, 5, 5) and («i,a

2,0-3) = (ej,£2,£3) • A, so 01 = hs

2 + 5f

3 , «2 = 5?i + 5c 3 , 03 = 5ei + 5

£2 • Then -5 5 5 Ol ' Oi

2 - y«i + -«2 + 7^3 and the module Z[ai, 02,03] is not. a ring, so ^[01,02,03] is not a semiorder. Example 2. Let £"1,^3, £3 and A be the same as in above example. Let 01 = 2t! , «2 = 2^2, 03 = 2e

3 . and (ßi,ß

2,ß

3) = («1,02,03) • A, so ßi - 5

O

2 + 5O

3 , ß

2 — 5a 1 + 5

O

3 , ß

3 = 5o 1 + 5a2 . Then ßl = -50a 1 - 100a

2 - 150a

3, ßl = -150oi - 50a

2 - 100a

3

, ßl = -100a! - 150a

2 - 50a

3

.