Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)

FREJMAN, D., Note on Abel's result about roots of polynomials

62 Dariusz Frejman where r

2's are distinct roots of Q(x). The proof of (3) easily follows by an algebraic method. oo Since F(x) = ^ ai

cx

m

k

, and 0 < m^ < n — 2, then k=o n jp/ \ n 1 oo Prom (4) we obtain n F(r ) n r

m° n r

m' n r

m

k (5

) Since 0 < rrik < n — 2 then by (3) and (5) we obtain yM = 0 Éí<3'(r0 and the proof of the Theorem is complete. Remark. Putting m^ = m — k, k = 0,1,..., ra; we have m m f{X) = Y,

a*

xmk = = where deg P{x) = m < n — 2, and we obtain (1). Rereferences [1] D. K. FADDIEÍEV and I. S. SOMINSKIÍ, Collection of the problems in higheralgebra, Moskov, 1964 (in Russian). [2] P. GRIFFITHS, "Variations on a theorem of Abel", Invent. Math. 35 (1976), 321-390. [3] M.S. GROSOF and G. TAIANI, "Vandermonde strikes again", Amer. Math. Monthly

, 1993, 575-577.