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

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