p-Rational Fields and the Structure of Some Modules

El, Habibi, Abdelaziz; Ziane, M'hammed

doi:10.48550/arxiv.1804.10165

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“…Of course, dim Fp (H 2 (G K,S , Z/pZ)), giving the minimal number of relations, is easily obtained only when P ⊆ S (equal to rk p (T K,S ) under Leopoldt's conjecture), which shall explain the forthcoming studies about this: [5] [NSW2000], [6,7] [Win1989-1991], [8] [Yam1993], [11] [Mai2005], [12] [Lab2006], [14] [Vog2007], [35] [Koch1970], [43] [Neu1976], Haberland [44] [Hab1978], [45] [Sch2010], El Habibi-Ziane [46] [ElHZ2018] . .…”

Section: 15mentioning

confidence: 92%

“…In [122, Table 1, § 2] [BR2017], Barbulescu-Ray give explicit p-rational large compositum of quadratic fields. We may cite some works by Bouazzaoui [123] [Bou2018], El Habibi-Ziane [46] [ElHZ2018] based on p-rationality of quadratic fields. In the similar context of p-ramification, a new PARI/GP pakage allows the computation of the logarithmic class group Cℓ K of a number field by Belabas-Jaulent [85] [BJ2016] that we can illustrate as follows where the invariants [X, Y, Z] are linked by the exact sequence: These are the only solutions for p < 10 8 .…”

Section: The Case Of Totally Real Fields Is Easier Since the Log-func...mentioning

confidence: 99%

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

Gras

2019

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The theory of p-ramification, regarding the Galois group of the maximal pro-p-extension of a number field K, unramified outside p and ∞, 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 ⊆ P , the structure of the Galois group of the maximal S-ramified abelian prop-extension HK,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 TK,S of Gal(HK,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.

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