= Iwasawa theorem on local cyclotomic units
{c}
{title2=$U_\infty^1/C_\infty^1\simeq\mathbb Z_p[[G]]/(I(G)\zeta_p)$}
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 $G=\mathbb Z_p^\times/\{\pm1\}$. 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.
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