COMPUTING NORMAL INTEGRAL BASES OF ABELIAN NUMBER FIELDS

Let L be an abelian number field of degree n with Galois group G. In this paper we study how to compute a normal integral basis for L, if there is at least one, assuming that the group G and an integral basis for L are known. The running time of the algorithm is dominated by the time required to compute the unit group of some cyclotomic fields and test whether some ideals are principal in these fields. When n is a prime power these two tasks can be accomplished quite efficiently thanks to recent results of Biasse, Fieker et al.