 Iwasawa theorem on local cyclotomic units

For the real p-power cyclotomic tower, take norm limits of local principal units and closed cyclotomic-unit subgroups. The corrected Coleman logarithm identifies $U_{\infty }^1$ with the Iwasawa algebra of $G={\mathbb{Z}}_p^{\times }/\{\pm 1\}$. On a symmetric cyclotomic unit it gives $([b]-[a]){\zeta }_p$. A topological generator generates the cyclotomic-unit limit and the augmentation ideal, yielding the quotient description. This local calculation does not alone identify the characteristic ideal of the global Galois module.