Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0584.12004
Gras, Georges
Plongements Kummeriens dans les ${\bbfZ}\sb p$-extensions. (Kummer embeddings in ${\bbfZ}\sb p$-extensions). (English)
[J] Compos. Math. 55, 383-396 (1985). ISSN 0010-437X; ISSN 1570-5846/e

Let k be a number field containing the group of $p\sp e$-th roots of unity (where p is a fixed prime and $e\ge 1)$. Let $\tilde k$ denote the composite of all ${\bbfZ}\sb p$-extensions of k; then the maximal subextension $\tilde N$ of $\tilde k$ of exponent $p\sp e$ is of the form $k(\sp{p\sp e}\sqrt{\tilde R})$. 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$\prod\sb{{\frak p}\in S}{\frak p}.$ \par In a preceding paper [J. Reine Angew. Math. 343, 64-80 (1983; Zbl 0501.12015)], the author defined a p-adic logarithm function Log: $I\to (\prod\sb{{\frak p}\in S}k\sb{{\frak p}})/\Lambda$, where $\Lambda$ is a ${\bbfQ}\sb p$-algebra generated by the image of units $\equiv 1$ mod$\prod\sb{{\frak p}\in S}{\frak 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. \par 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 $p\sp e=2$ 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.
[T.Takeuchi]

MSC 2000:: *11R18 Cyclotomic extensions
11S15 Ramification and extension theory
11S80 Other analytic theory of local fields
11R23 Iwasawa theory

Keywords: maximal Kummerian subextension; ${\bbfZ}\sb p$-extensions; p-adic logarithm; Artin group; set of generators; p-power residue symbol; Kummer theory; generators of the 2-class group; examples

Citations: Zbl 0509.12008; Zbl 0501.12015

Cited in: Zbl 0635.12002

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]