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The theory of p-ramification, regarding the Galois group of the maximalpro-p-extension of a number field K, unramified outside p and $\infty$, is wellknown including numerical experiments with PARI/GP programs. The case of``incomplete p-ramification'' (i.e., when the set S of ramified places is astrict subset of the set P of the p-places) is, on the contrary, mostly unknownin a theoretical point of view. We give, in a first part, a way to compute, forany S $\le$ P, the structure of the Galois group of the maximal S-ramifiedabelian pro-p-extension H\_(K,S) of any field K given by means of anirreducible polynomial. We publish PARI/GP programs usable without any specialprerequisites. Then, in an Appendix, we recall the ``story'' of abelianS-ramification restricting ourselves to elementary aspects in order to precisemuch basic contributions and references, often disregarded, which may be usedby specialists of other domains of number theory. Indeed, the torsion T\_(K,S)of 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, Galoiscohomology, p-adic L-functions, elliptic curves, algebraic geometry, wouldmerit special developments, which is not the purpose of this text.

Practice of incomplete p-ramification over a number field -- History of abelian p-ramification