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

## TL;DR

This paper explores incomplete p-ramification in number fields, providing computational methods and PARI/GP programs to analyze the Galois group structure of maximal S-ramified abelian pro-p-extensions, and reviews foundational aspects of abelian S-ramification.

## Contribution

It introduces a computational approach to determine the Galois group structure for any subset of p-places and provides accessible PARI/GP programs, expanding understanding of incomplete p-ramification.

## Key findings

- Provides a method to compute Galois groups for S-ramification
- Publishes PARI/GP programs for practical computations
- Reviews foundational aspects of abelian S-ramification

## Abstract

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 $\le$ 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 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, Galois cohomology, p-adic L-functions, elliptic curves, algebraic geometry, would merit special developments, which is not the purpose of this text.