The Galois groups of the polynomials $x^n+ax^s+b$, II
Hiroyuki Osada · Tohoku Mathematical Journal · 1987
Introduction.In the previous paper [3], we have shown that the Galois group of a polynomial fix) = x n + ax 8 + b (with rational integers a and b) over the rational number field Q is isomorphic to the symmetric group S n of degree n under the following conditions:(1) f(x) is irreducible over Q.(2) a = a Q c n , b = 6 0 c n and (a o c(n -s)s, nb 0 ) = 1 (relatively prime).(3) |A(/)I is not a square, where(5) There exists a prime number q such that g|s and k < q for any positive integer k with k and & < s/2.In this paper, we shall first show that the same result holds without the assumption (5) (Theorem 1).Further, we shall show that there exist infinitely many polynomials x n + ax 8 + p satisfying the above conditions (1), (2), ( 3) and (4) (Theorem 2).By Hubert's irreducibility theorem [2], there exist infinitely many Galois extensions with Galois group S n or A n for any n.Schur [4, gave a criterion for the Galois group of a polynomial over Q to be isomorphic to S n or to A n .We here give another criterion for the Galois group of a polynomial over Q to be isomorphic to S n or to A n (Theorem 3).As another consequence of our results, we can also construct infinitely many polynomials with the Galois groups A i9 A Q and A 7 (Corollary 3, Corollary 4 to Theorem 3 and Proposition 2).Besides, we give numerical examples of polynomials with Galois group A 7 .The author would like to thank the referee for his valuable advices.Let Z be the ring of rational integers.Throughout this paper, we shall denote by K, G and D(f) the splitting field, the Galois group and the discriminant of a polynomial f(x)eZ[x], respectively.