Practice of the Incomplete $p$-Ramification Over a Number Field -- History of Abelian $p$-Ramification

The theory of $p$-ramification, regarding the Galois group of the maximal pro-$p$-extension of a number field $K$, unramified outside $p$ and $\infty$, is well known including numerical experiments with PARI/GP programs. The case of ``incomplete $p$-ramification'' (i.e., when the set $S$ of ramified places is a strict subset of the set $P$ of the $p$-places) is, on the contrary, mostly unknown in a theoretical point of view. We give, in a first part, a way to compute, for any $S \subseteq P$, the structure of the Galois group of the maximal $S$-ramified abelian pro-$p$-extension $H_{K,S}$ of any field $K$ given by means of an irreducible polynomial. We publish PARI/GP programs usable without any special prerequisites. Then, in an Appendix, we recall the ``story'' of abelian $S$-ramification restricting ourselves to elementary aspects in order to precise much basic contributions and references, often disregarded, which may be used by specialists of other domains of number theory. Indeed, the torsion ${\mathcal T}_{K,S}$ of ${\rm Gal}(H_{K,S}/K)$ (even if $S=P$) is a fundamental obstruction in many problems. All relationships involving $S$-ramification, as Iwasawa's theory, Galois cohomology, $p$-adic $L$-functions, elliptic curves, algebraic geometry, would merit special developments, which is not the purpose of this text.

Keywords:

Abelian $S$-ramification, Class field theory, Class groups, Leopoldt's conjecture, $p$-adic regulators, Pro-$p$-groups, Units $\Z_p$-extensions,

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