Rank functions, Zₚ-extensions and class numbers

If F is a number field and F = F₀ ⊂ F₁ ⊂ F₂⊂ ⋯ is a tower of Galois extensions with Gal(Fₙ/F) cyclic of order pⁿ, K. Iwasawa gave a closed formula for the growth of the p-part of the class number of Fₙ whenever n is large enough. We present a new proof of this formula, obtained by reinterpreting its summands in terms of Sylvester matrix rank functions on the Iwasawa algebra $Λ$ without resorting to the classification of finitely-generated $Λ$-modules up to pseudo-isomorphism.