On the common index divisors of an algebraic field

whose coefficients are rational integers. We shall call (1) the characteristic equation for 0. If do is the discriminant of 0 then do= ko2 d where ko is a rational integer, the of 0. A divisor to the indices of every integer of the field has been called by Kronecker a gemeinsamer ausserwesentlicher Discriminantenteiler of the field. We shall use the term common divisor.t The existence of divisors was first established in 1871 by Dedekindt who exhibited examples in fields of third and fourth degrees. Dedekind? further showed that a rational prime p can be a divisor of a field K if and only if at least one of the inequalities r(f) >g(f) holds, where r(f) is the number of prime ideal divisors of p of degree f, and g(f) is the number of different prime functions (mod p) of fth degree. Using Kronecker's theory of algebraic numbers, Henselll has given a necessary and sufficient condition on the so-called index form for p to be a divisor. In 1907, Bauer? showed that if p <n there exists a field of nth degree in which p is a divisor. Von Zylinsky** has