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

Note on Abel's result about roots of polynomials DARIUSZ FREJMAN Abstract. In this Note we give an algebraic proof of the well-know Abel's result about roots of polynomials. Our proof is other than the proof given by M. S. Grosof and G. Taiani in the paper [3]. A theorem of Abel [2] states: if P(x

), Q(x) are any polynomials such that deg Q = n > 3,Q(x) has no multiple roots and deg P — m < n — 2, then (i) y£M = 0 where r

t-'s are distinct roots of Q(x) and Q'(x) denotes the derivate of Q(x). We remark that the original proof given by Abel uses integrals. The modern proof can be based on residue theory. In the paper [3] given by M. S. Grosof and G. Taiani has been presented an algebraic proof of (1) in spirit of classical theory of equations by using some property of the Vandermonde's determinant. The purpose of this note is to present also algebraic proof of (1) by different method. Nemely, we prove the following: oo Theorem. Let F(x) = ^ akX

m

k

, where 0 < m^ < n — 2 and Q(x) be the k—0 polynomial of the degree n > 3, such that Q(x) has no multiple roots. Then where r^s are distinct roots of Q(x) and Q'(x

) denotes the derivate of Q(x). Proof. In the proof of (2) we use of the following result (see [1], p. 87 and solution on p. 220). If Q(x

) has no multiple roots and deg<5(^) = n > 3, then for every natural fixed s, such that 0<s<n — 2, we have; (3

) =