Let D be the discriminant of an algebraic number field F of degree n over the rational field R. The problem of finding the lowest absolute value of D as F varies over all fields of degree n with a given number of real (and consequently of imaginary) conjugate fields has not yet been solved in general. The only precise results so far given are those for n = 2, 3 and 4. The case n = 2 is trivial; n = 3 was solved in 1896 by Furtwangler, and n = 4 in 1929 by J. Mayer [6]. Reference to Furtwangler's work is given hi Mayer's paper. In this paper the results for n = 5, that is, for quintic fields, are obtained.

The purpose of this paper is to prove a theorem which concerns the normal subgroup of a free product Π generated by a given subset Ω. This theorem was stated in the first paper of this series (Britton [1]) and an application was made to the word problem. The present work is, however, independent.

Airy's integrals

can be expressed ([1], [2]) in terms of Bessel functions. In this paper integrals of the types

are discussed. Various subsidiary formulae are given in § 2, some integrals of the type

are evaluated in § 3, and from these the integrals of the Airy type are derived in § 4.

In integrating E-functions with respect to their parameters the contours are usually of the Barnes type, deformed if necessary to separate the increasing and decreasing sequences of poles of the integrands. Also the constants are taken to be such that the integrals converge. The following formulae are required in proving the theorems given in this paper.

The formula

was stated by Bateman ([2], p. 457); a proof is sketched in [3], p. 144. Here

the Laguerre polynomial of degree

Let

be the rath cyclotomic polynomial, and denote by An the absolute value of the largest coefficient of Fn(x).Schur proved that

and Emma Lehmer [5] showed that An>cn1/3 for infinitely many n; in fact she proved that n can be chosen as the product of three distinct primes. I proved [3] that there exists a positive constant q such that, for infinitely many n

and Bateman [1] proved very simply that, for every ∈>0 and all n>no(∈),