Computation of the $p$-part of the ideal class group of certain real abelian fields

Abstract

Under Greenberg's conjecture, we give an efficient method to compute the p-part of the ideal class group of certain real abelian fields by using cyclotomic units, Gauss sums and prime numbers. As numerical examples, we compute the p-part of the ideal class group of the maximal real subfield of Q(√-f, ζ p n+1) in the range 1 < f < 200 and 5 ≤ p < 100000. In order to explain our method, we show an example whose ideal class group is not cyclic.