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

4 P. G. Tsangaris f Fn/^x

8)^!^!^ for n > 1 except for d = n = 2, R

v(v

s - i»

s, F

n{v)) = i -F^x

3) = 1 - for d = n = 2, i (-l)

i+

1Fi(x

s) = - 1) for n = 1. Proof. Let R(x) = R

v {v

s-x

s, F

n(v)), G{x) = F

n/

d(a:')*W/*(»/<0 and />

1

? p

2,..., p

s be the s-th roots of unity. Then p\X, p

2x,..., p

sx are the roots of v

s — x

s (for x fixed). Hence R(x) = F

n(pix) • • • F

n(p

sx). Let £ be a root of R(x). Hence, F

n(pk£) — 0 for some k

, with 1 < k < s, i.e. pk£ is a root of F

n(v). Thus, pk£ is a primitive n-th root of unity. Set pki — C> then = (,*

s

. But the order of £

3 is n/d. Hence is a primitive n/d-th root of unity, i.e. Hence, F

n/

d(C)

,p{n)/4>(n,d

) = 0, i.e. £ is a root of G(x). Hence, every root of R(x) is a root of i.e. R(x)\G(x). (1) Also degG(z) = deg R(x) = s<f>(n). (2) From (1) and (2) we have: G(x) = cR(x), where c is a (rational) constant. (3) Hence G(0) = ci?(0), that is F

n/d{ ())*<")/*(»/*) = cF

n(0)

5. (4) To derive the sought formula it suffices now to evaluate the constant c. We have to examine two cases: (a) If n > 1. In case d / n, then n/d > 1. Also F

n(

0) = 1 and Fi(0) = -1. Then, in view of (4) we have c = 1. In case d = n > 1, we have in view of (4) that 1 ' - \ 1, if n > 2.