ON $ p$-CLOSED ALGEBRAIC NUMBER FIELDS WITH RESTRICTED RAMIFICATION

Neumann, Olaf

doi:10.1070/im1975v009n02abeh001475

Cited by 9 publications

(5 citation statements)

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“…Example with p totally split in degree 7. For the polynomial P = x 7 − 7 and p = 43, one finds two cases: [43] i.e., r K,S = 0 and T K,S ≃ Z/43Z for the two above cases. For the other modulus, T K,S = 1.…”

Section: Algorithmic Approach Of S-ramificationmentioning

confidence: 97%

“…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: 99%

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

Gras

2019

Preprint

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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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Section: Algorithmic Approach Of S-ramificationmentioning

confidence: 97%

Section: 15mentioning

confidence: 99%

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

Gras

2019

Preprint

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show abstract

“…Of course, dim F p (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], [6,7], [8], [11], [12], [14], [36], [44], Haberland [45], [46], El Habibi-Ziane [47] . .…”

Section: A21šafarevič Formulamentioning

confidence: 99%

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

Gras¹

2019

Communications in Advanced Mathematical Sciences

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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 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. Practice of the Incomplete p-Ramification Over a Number Field -History of Abelian p-Ramification -252/280The analogous theory for a local base field has also a long history that we shall not consider in this article.

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“…(Neumann (1975), Haberland (1978), Proposition 22) lEI By means of Theorem 3.73 one transfers the results about the cohomology of Gs to Gs(p): We put Vs(K):= {IX E K X IIX E UvK;P for all finite places v, IX E K;P for v E S}, is an isomorphism for all n ~ O.…”

Section: Also As Generators Of G(ldk)mentioning

confidence: 99%

Algebraic Number Theory

Koch¹

1997

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ON $ p$-CLOSED ALGEBRAIC NUMBER FIELDS WITH RESTRICTED RAMIFICATION

Cited by 9 publications

References 3 publications

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

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

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

Algebraic Number Theory

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