DIHEDRAL QUINTIC POLYNOMIALS AND A THEOREM OF GALOIS

ABSTRACT. Let r1 and r2 be any two roots of a monic irreducible quintic polynomial in Q[X] with Galois group D5. It is shown how to determine the other three roots as rational functions of r1 and r2 in accordance with a theorem of Galois. 1. Introduction If f(X) ∈ Q[X] is a solvable irreducible polynomial of prime degree, a theorem of Galois asserts that all the roots of f(X) can be given as rational functions of any two of them. When the degree of f(X) is 2 or 3 this assertion is trivial. However it is an unsolved problem to determine these rational functions when f(X) is of degree 5 [1, p. 355]. In this paper we show how to find these rationa