The distribution of prime ideals in a real quadratic field with units having a given index in the residue class field

Abstract

Let k be a real quadratic number field and o k, E the ring of integers and the group of units in k. Denote by E p the subgroup represented by elements of E of ( o k/ p) × for a prime ideal p in k. We show that for a given positive rational integer a, the set of prime numbers p for which the residual index of E p for p lying above p is equal to a has a natural density c under the Generalized Riemann Hypothesis. Moreover, we give the explicit formula of c and conditions to c=0.