Cyclotomic unit index formula ($v_p[{\mathcal{O}}_{K_n^{+}}^{\times }:D_n]=v_p(h_{K_n^{+}})$)

For a real cyclotomic field of odd prime-power conductor, the index of its cyclotomic-unit group has the same p-part as its ideal class number. The analytic class number formula compares the regulator of all units with the explicit cyclotomic-unit regulator. The equality here is stated at the p-part level, so harmless prime-to-p normalization factors do not affect it.