Structure and prime decomposition law and relative extensions of abelian fields with prime power degree

Abstract

LetL be an abelian extension of the rationals Q whose Galois group Gal(L) is an abelian (q-groupq is any prime number). The explicit law of prime decomposition inL for any prime numberp, the inertia group, residue class degree, and discriminant ofL are given here; such fieldsL are classified into 4 or 8 classes according asq is odd or even with clear description of their structures. Then relative extensionL/K is studied.L/K is proved to have a relative integral basis under certain simple conditions; relative discriminantD(L/K) is given explicitly; and necessary and sufficient conditions are obtained forD(L/K) to be generated by a rational square (and by a rational). In particular, it is proved thatL/K has a relative integral basis and thatD(L/K) is generated by a rational square if [L: K] ⩽x* orx * +1 (according asq is odd or even), where x* is the exponent of Gal(L). These results contain many related results on similar fields in literature.