On the number of real roots of a solvable polynomial

Abstract

1. Introduction and Loewy’s theorem. By a classical theorem the number of real roots of an irreducible polynomial f(X) of odd prime degree p over a real number eld K is either 1 or p if the Galois group of f(X) over K is solvable. This result was generalized by A. Loewy in the following way: For a polynomial f(X) we let r(f) denote the number of real roots of f(X). Loewy’s theorem. Let K be a real number eld and f(X) an irreducible polynomial in K[X] of odd degree n. If p is the smallest prime divisor of n and the Galois group of f(X) over K is solvable, then r(f) = 1 or n or satises the inequalities p r(f) n p + 1. When the degree of f(X) is a prime number the above theorem is an immediate corollary to the following Galois’ theorem. Let f(X) be an irreducible separable polynomial over a eld K having a solvable Galois group over K. If the degree of f(X) is a prime number, then any two roots of f(X) generate the splitting eld of f(X) over K. Galois’ theorem, which is basically a group-theoretic result, cannot be generalized to yield a proof of Loewy’s theorem. Indeed, for any odd prime number p and any t, 1 t p, there exists an irreducible polynomial f(X) in Q[X] of degree p 2 with solvable Galois group having t roots 1;:::; t