Existence of Integral Bases for Relative Extensions ofn-Cyclic Number Fields Original Research Article

Suppose thatLsuperset ofKare abelian extensions of the rationalsQwith Galois groups (Z/qsZ)nand (Z/qrZ)m, respectively,qany prime number. It is proved thatL/Khas a relative integral basis under certain simple conditions. In particular, [L:K]greater-or-equal, slantedqsorqs+1 (according toqis odd or even) is enough. The relative discriminantD(L/K) is also computed explicitly and is proved to be generated by a rational square under a (necessary and sufficient) condition