Let \(K\) be an algebraic number field with ring of integers \({\mathcal O}_K\), and \(b\) be an odd prime. The number of independent \(\mathbb{Z}_p\)-extensions of \(K\) is conjectured by Leopoldt to be \(r_2+1\), where \(r_2\) is the number of complex places of \(K\). This conjecture can be placed in the context of Galois extensions of commutative rings as follows: Let \(R={\mathcal O}_K[1/p]\), then a \(\mathbb{Z}_p\)-extension of \(K\) is unramified except at primes lying over \(p\), hence gives rise to a tower of \(C_n\)-Galois extensions of \(R\) for \(n>0\), \(C_n=\) cyclic group of order \(p^n\). Let \(\mathrm{Gal}(R,C_n)\) be the group of (isomorphism classes of) Galois extensions of \(R\) with Galois group \(C_n\), and \(\mathrm{Gal}(R,\mathbb{Z}_ p)=\underleftarrow{\lim}\,\mathrm{Gal}(R,C_n)\): then Leopoldt’s conjecture is that \(\mathrm{Gal}(R,\mathbb{Z}_p)\cong\mathbb{Z}_p^{r_2+1}\). Let \(NB(R,C_n)\) denote the subgroup of Galois extensions with normal basis, and \(NB(R,\mathbb{Z}_p)=\underleftarrow{\lim}\,NB(R,C_n)\). The author proves that \(NB(R,\mathbb{Z}_p)\cong\mathbb{Z}_p^{r_2+1}\). This result was proved by I. Kersten and J. Michalicek [J. Number Theory 32, 131–150 (1989; Zbl 0709.11057)] if \(K\) is totally real or a \(CM\) field.
To obtain these results, the first part of the paper is devoted to understanding \(\mathrm{Gal}(R,C_n)\) when \(R\) is a connected commutative ring containing \(1/p\). If \(R\) also contains a primitive \(p^n\)-th root of unity \(\zeta\), then there is an exact Kummer sequence:
\[ 0\to NB(R,C_n)\to\mathrm{Gal}(R,C_n)\to{_n\mathrm{Pic}(R)}\to 0 \]
where \(NB(R,C_n)\cong U(R)/U(R)^{p^n}\) and \(_n\mathrm{Pic}(R)\) is the \(p^n\)-torsion subgroup of \(\mathrm{Pic}(R)\) [see the reviewer, Ill. J. Math. 15, 273–280 (1971; Zbl 0211.37102) or A. Z. Borevich, J. Sov. Math. 11, 514–534 (1979), transl. from Zap. Nauchn. Semin. Leningr., Otd. Mat. Inst. Steklova 57, 8–30 (1976; Zbl 0379.13003)]. The main technique for dealing with \(\mathrm{Gal}(R,C_n)\) when \(\zeta\notin R\) is descent. Let \(S=R[\zeta]\), then \(S\) is a Galois extension of \(R\) with group \(\Gamma\cong U(\mathbb{Z}/p^n\mathbb{Z})\). The natural action of \(\Gamma\) on \(\mathrm{Gal}(S,C_n)\) and a twisted action \(t\) of \(\Gamma\) on the other groups in the Kummer sequence over \(S\), yield the desired short exact sequence for \(\mathrm{Gal}(R,C_n)\):
\[ 0\to NB(R,C_ n)\to\hbox{Gal}(R,C_ n)\to P\to 0 \]
where \(P\cong(_n\mathrm{Pic}(S))^{t\Gamma}\) and \(NB(R,C_n)\cong(U(S)/U(S)^n)^{t\Gamma}\).