On abelian fields

By Kronecker'sJ Theorem on Abelian fields, all such fields are subfields of cyclotomic fields, that is, fields generated by a root of unity. Abelian fields may then be classified by considering all cyclotomic fields and sorting the subfields in some manner that will exclude repetition. For example this is done, in part at least, by Weber by making use of the notion of primary subfields: a subfield of Qm, the field generated by a primitive mth root of unity, is a primary subfield if it is not contained in an flm. (m' <m). We here make use of what we shall call simple^ (primary) subfields as defined below. If then the (known) discriminants of Abelian fields are set up on this basis, a number of properties of Abelian fields become apparent. In particular is this true of the fields contained in a fixed simple subfield (see §5). In §6 some results on common index divisors (that is, common inessential discriminantal divisors) are obtained. Using a necessary and sufficient condition valid for any algebraic field it is shown how to derive for the case of Abelian fields very simple criteria that a given rational prime be a common index divisor. The criteria are of two kinds. A typical instance of the first kind is the following. Let q and / be odd primes such that 1 = 1 (mod q); let C denote that cyclic subfield of ñ¡ that is of degree q. Then a necessary and sufficient condition that a prime p(p<q) be a common index divisor of C is that