Plongements Kummeriens dans les \({\mathbb{Z}}_ p\)-extensions. (Kummer embeddings in \({\mathbb{Z}}_ p\)-extensions). (English) Zbl 0584.12004

Let k be a number field containing the group of \(p^ e\)-th roots of unity (where p is a fixed prime and \(e\geq 1)\). Let \(\tilde k\) denote the composite of all \({\mathbb{Z}}_ p\)-extensions of k; then the maximal subextension \(\tilde N\) of \(\tilde k\) of exponent \(p^ e\) is of the form \(k(^{p^ e}\sqrt{\tilde R})\). Let S be the set of primes of k lying above p, and I the group of ideals of k of which a p-power is generated by an element \(\equiv 1\) mod\(\prod_{{\mathfrak p}\in S}{\mathfrak p}.\)
In a preceding paper [J. Reine Angew. Math. 343, 64-80 (1983; Zbl 0501.12015)], the author defined a p-adic logarithm function Log: \(I\to (\prod_{{\mathfrak p}\in S}k_{{\mathfrak p}})/\Lambda\), where \(\Lambda\) is a \({\mathbb{Q}}_ p\)-algebra generated by the image of units \(\equiv 1\) mod\(\prod_{{\mathfrak p}\in S}{\mathfrak p} \); and, for a p-ramified finite p- abelian extension M of k, gave an expression of the Artin group of \(M\cap \tilde k\) using Log.
In the present paper, using this result, the author gives an algorithm to calculate a set of generators of \(\tilde R\) from the p-class group and units of k which consists of the determination of the Artin group \(\tilde A\) of \(\tilde N\) by calculating Log, and the computation of \(\tilde R\) from \(\tilde A\) using the p-power residue symbol through Kummer theory. Especially, in the case that \(p^ e=2\) and k is imaginary quadratic, he computes explicitly several steps of this, and obtains an effective and systematic method to calculate \(\tilde R\) from a set of generators of the 2-class group of k. Several examples are also given.
Reviewer: T.Takeuchi


11R18 Cyclotomic extensions
11S15 Ramification and extension theory
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11R23 Iwasawa theory
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