On vandiver’s conjecture

Jakubec, Stanislav

doi:10.1007/bf02940778

Cited by 7 publications

(4 citation statements)

References 9 publications

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“…Let F n _j be an (n -1 )-dimensional vector space over the field Z/pZ. Define a mapping Y K : U K^Vn _ x by LEMMA 1 OF [6]. The mapping *F K is a homomorphism,:…”

Section: Consider the Polynomial/(x) = A O + A 1 X+ ••• +A N X N Dementioning

confidence: 99%

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Congruence of Ankeny–Artin–Chowla type for cyclic fields of prime degree l

Jakubec¹

1996

Math. Proc. Camb. Phil. Soc.

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“…Let F n _j be an (n -1 )-dimensional vector space over the field Z/pZ. Define a mapping Y K : U K^Vn _ x by LEMMA 1 OF [6]. The mapping *F K is a homomorphism,:…”

Section: Consider the Polynomial/(x) = A O + A 1 X+ ••• +A N X N Dementioning

confidence: 99%

“…Let if be a cyclic cubic field of prime conductor p, p = 1 (mod3). Let The proof of the theorem is based on results of the paper [6]. Now we shall introduce some facts necessary for the proof.…”

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confidence: 99%

Congruence of Ankeny–Artin–Chowla type for cyclic fields of prime degree l

Jakubec¹

1996

Math. Proc. Camb. Phil. Soc.

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“…In a sequence of papers [J1,J2,J3,J4,J5,J6,JL], Stanislav Jakubec has derived a purely algebraic technique that enabled him to prove congruences of Ankeny-Artin-Chowla type modulo p and p 2 for cyclic fields K of prime degree l and prime conductor p. For an overview of his technique we recommend reading the first section of [M]. His main result (see also [M,Theorem 1.1]) is a congruence for the class number h K of the field K expressed using certain units of finite index in terms of a technical map Φ n defined later.…”

Section: Introduction and Notationmentioning

confidence: 99%

Decomposition of congruences involving a map Φ

Marko

2010

Mathematica Slovaca

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Congruences of Ankeny-Artin-Chowla type modulo p 2 for a cyclic subfield K of prime conductor p were derived by Jakubec and expressed in terms of a technically defined map Φ. Later, Jakubec and Lassak found a decomposition of the map Φ modulo p 2 and simplified the formulation of these congruences. A corresponding decomposition of the map Φ modulo p 3 was obtained in [MARKO, F.: Towards Ankeny-Artin-Chowla type congruence modulo p 3, Ann. Math. Sil. 20 (2006), 31–55]. That technical step was important for the formulation of congruences of Ankeny-Artin-Chowla type modulo p 3. This paper will show how to produce an analogous decomposition of the map Φ modulo an arbitrary power p n which would allow a description of analogous congruences modulo p n.

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“…For the history of the above congruence and conjecture see [16]. All of this is quite important but we would like to look for explicit congruences of the same type that are valid for a different class of fields and modulo higher powers of p. In this direction, S. Jakubec, using algebraic techniques in [3][4][5][6][7][8][9], has established congruences of Ankeny-Artin-Chowla type modulo p and p 2 for totally real cyclic fields K of degree l and prime conductor p. Perhaps his technique is best summarized in [14: Section 1], the notation of which we are going to follow (reader is encouraged to consult [14] if needed). Since congruences modulo p and p 2 were previously derived by Jakubec, it is natural to investigate the next case of congruences modulo p 3 , hoping to gain insight into the general case modulo higher powers of p.…”

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confidence: 99%