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A compact Galois module is a compact profinite abelian group equipped with a continuous Galois group action. An abelian pro-p group is naturally a compact ${\mathbb{Z}}_p$-module. A continuous action of the Galois group of a Zp-extension extends to its Iwasawa algebra of a Zp-extension. This compact topology differs from the discrete topology usually used in Galois cohomology.

For a compact continuous module over the Iwasawa algebra of a Zp-extension, lifts of a basis of $M/(p,T)M$ generate $M$. Their generated image is compact. The quotient $Q$ satisfies $Q=(p,T)Q$; in every finite continuous quotient, $(p,T)$ acts nilpotently, since the group action is pro-$p$. Thus all finite quotients of $Q$ vanish, and compactness gives $Q=0$.

The unramified Iwasawa module is finitely generated torsion over the Iwasawa algebra of a Zp-extension, for every Zp-extension of a number field. After a finite shift all ramified primes are totally ramified and their number $s$ is constant. Class field theory bounds finite-layer coinvariant modules by ${\mathrm{rank}}_{{\mathbb{Z}}_p}{Y}_{{\Gamma }_n}\le s-1$. The Compact Nakayama lemma proves finite generation, and a positive Iwasawa-module rank would force ranks at least $r{p}^n$, a contradiction. No Leopoldt conjecture is required.