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

Georges GRAS

doi:https://doi.org/10.33434/cams.573729

Article Details

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

Authors:

Georges GRAS

Publisher:

Emrah Evren KARA

Main Topics:

Mathematics

Journal:

Communications in Advanced Mathematical Sciences 2019 Vol.2 Issue 4, pp.251-280

Document Type:

Article

Publication Date:

12/2019

Document Language:

English

Document ID (Org No.):

2000911686

Frequency:

Quarterly

pISSN:

2651-4001

DOI:

https://doi.org/10.33434/cams.573729

Abstract:

 

The theory of ppp-ramification, regarding the Galois group of the maximal pro-ppp-extension of a number field KKK, unramified outside ppp and ∞∞\infty, is well known including numerical experiments with PARI/GP programs. The case of incomplete ppp-ramification'' (i.e., when the set SSS of ramified places is a strict subset of the set PPP of the ppp-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⊆PS⊆PS \subseteq P, the structure of the Galois group of the maximal SSS-ramified abelian pro-ppp-extension HK,SHK,SH_{K,S} of any field KKK 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 SSS-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 TK,STK,S{\mathcal T}_{K,S} of Gal(HK,S/K)Gal(HK,S/K){\rm Gal}(H_{K,S}/K) (even if S=PS=PS=P) is a fundamental obstruction in many problems. All relationships involving SSS-ramification, as Iwasawa's theory, Galois cohomology, ppp-adic LLL-functions, elliptic curves, algebraic geometry, would merit special developments, which is not the purpose of this text.

Article Details

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

Authors: Georges GRAS

Publisher: Emrah Evren KARA

Main Topics: Mathematics

Journal: Communications in Advanced Mathematical Sciences 2019 Vol.2 Issue 4, pp.251-280

Document Type: Article

Publication Date: 12/2019

Document Language: English

Document ID (Org No.): 2000911686

Frequency: Quarterly

pISSN: 2651-4001

DOI: https://doi.org/10.33434/cams.573729

Issues

Authors:

Georges GRAS

Publisher:

Emrah Evren KARA

Main Topics:

Mathematics

Journal:

Communications in Advanced Mathematical Sciences 2019 Vol.2 Issue 4, pp.251-280

Document Type:

Article

Publication Date:

12/2019

Document Language:

English

Document ID (Org No.):

2000911686

Frequency:

Quarterly

pISSN:

2651-4001

DOI:

https://doi.org/10.33434/cams.573729