Circular units in the \(\mathbb Z_p\)-extensions of real abelian fields of prime conductor
Let \(k\) be a real abelian field and \(k_\infty\) its cyclotomic \(\mathbb Z_p\)-extension, \(p\) being a prime prime to the degree \([k:\mathbb Q]\). Let \(G_{m,n}\) be the Galois group between two different layers \(k_n\) and \(k_m\) in \(k_\infty (n\leq m)\). The paper computes the Galois cohomology of the Sinnott group of cyclotomic units \(C_m\) in \(k_m\): 1) \(H^0(G_{m,n},C_m)=C_n\), 2) \(\widehat H^0(G_{m,n},C_m)\simeq (\mathbb Z_p/p^{m-n}\mathbb Z_p)^{l-1}\), 3) \(\widehat H^1(G_{m,n},C_m)\simeq (\mathbb Z_p/p^{m-n}\mathbb Z_p)^{l}\) The main tool is \textit{V. Ennola}'s work on relations of cyclotomic units [J. Number Theory 4, 236--247 (1972; Zbl 0236.12005)].
scientific article; zbMATH DE number 1389722