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

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AbstractLet k be a real quadratic number field and ok,E the ring of integers and the group of units in k. Denote by Ep the subgroup represented by elements of E of (ok/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 Ep 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.

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