Abstract. In this paper and a forthcoming joint one with Y. Hachimori [15] we study Iwasawa modules over an infinite Galois extension k ∞ of a number field k whose Galois group G = G(k ∞ /k) is isomorphic to the semidirect product of two copies of the p-adic integers Z p . After first analyzing some general algebraic properties of the corresponding Iwasawa algebra, we apply these results to the Galois group X nr of the p-Hilbert class field over k ∞ . The main tool we use is a version of the Weierstrass preparation theorem, which we prove for certain skew power series with coefficients in a not necessarily commutative local ring. One striking result in our work is the discovery of the abundance of faithful torsion Λ(G)-modules, i.e. non-trivial torsion modules whose global annihilator ideal is zero. Finally we show that the completed group algebra F p [[G]] with coefficients in the finite field F p is a unique factorization domain in the sense of Chatters [8].
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.