We study the problem of constructing and enumerating, for any integers $m,n\>1$, number fields of degree $n$ whose ideal class groups have “large" $m$-rank. Our technique relies fundamentally on Hilbert’s irreducibility theorem and results on integral points of bounded degree on curves.

We define a sequence of rational integers $u_i\left(E\right)$ for each finite index subgroup E of the group of units in some finite Galois number fields K in which prime p ramifies. For two subgroups E’ ⊂ E of finite index in the group of units of K we prove the formula $v_p\left([E:E^{\text{'}}]\right)={\sum }_{i=1}^r{u}_i\left(E^{\text{'}}\right)-u_i\left(E\right)$. This is a generalization of results of P. Dénes [3], [4] and F. Kurihara [5].

The problem of determining power integral bases in algebraic number fields is equivalent to solving the corresponding index form equations. As is known (cf. Győry [25]), every index form equation can be reduced to an equation system consisting of unit equations in two variables over the normal closure of the original field. However, the unit rank of the normal closure is usually too large for practical use. In a recent paper Győry [27] succeeded in reducing index form equations to systems of unit...