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

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 $\mathbf {Q}(\sqrt {-f},\zeta _{p^{n+1}})$ in the range $1 <f<200$ and $5 \le p <100000$. In order to explain our method, we show an example whose ideal class group is not cyclic.