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

Tsangaris, P. G., Prime numbers and cyclotomy

Prime numbers and cyclotomy 7 (b) If n = 1, then (4) implies that {

1, if s is odd, — 1, if s is even. Remark. Theorem 1.2 should be considered as closely related to a corresponding Theorem of T. Apostol [1] on the resultant of the cyclotomic polynomials F

m(ax) and F

n(bx). Theorem 1.2. Let n,s be nat ural numbers. Denote by pi = I, pi,

..., p

s all the s-th roots of unity, and Jet An (x) = F

n(pix)

' •' F

n(p

s x) - F

n{pi) • • • F

n (p

s). Then: (i) (x

s-l)\f^(x). (ii) If n J(s, then (i-cr^inW"!.« where L°

n(x) = K°

n(x)/(x° Proof. The numbers pi, p2,

..., p

s form a cyclic group. Hence A'n [pk

) = F

n (pi p

k)--- F

n (p

s p

k) - F

n (pi) • • • F

n(p

s) = 0 for k = 1,2,..., s. Also pix,

..

., p

sx are the roots of v

s — x

s = (J (for x fixed). Thus K°

n(x) = R

v(v

s - x*,F

n(v)) - R(v

s - 1, F

n(v)) is a polynomial of x with integer coefficients. Since every pk is a root, of K^(x), part (i) follows immediately. Then i* (0 = /C'(C)/(C* - 1) and so K(0 = —Fn(pi) • • • F

n(p

s) = -R(o

$ - l,F

n(v)). In conclusion (1 - C

5)1 = L'

n{C)/R(v' - 1), F

n(v)). Theorem 1.3. Let. n, k be natural numbers such that n > 1, n j(k and let d = (n

, k). Define K*(x) = F

n/

d{x

k)'

t,(n)/(t,(n/d

) - F

n/

d(\)<)>(n)lHnld).