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The Genesis of a Theorem in the Galois Theory of p-Extensions of ℚ with Restricted Tame Ramification

Received: 26 July 2024     Accepted: 20 August 2024     Published: 26 August 2024
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Abstract

This article traces the genesis of a theorem that gives for the first time examples of the Galois group GS of the maximal p-extension of ℚ, unramified outside a finite set of primes not containing an odd p, that are of cohomlogical dimension 2 if the primes in S satisfy a certain linking condition. Because the ramification is tame the pro-p-group GS has all of its derived factors finite which is a strong finitenesss condition on GS. The paper starts with a question of Serre on one relator pro-p-groups and then a detour to discrete groups where the notion of strong freeness for a sequence of homogeneous Lie elements is given and a criterion for strong freeness is established. These notions are then carried over to pro-p-groups where the linking condtion on the primes of S is translated into a cohomological criterion for a pro-p-group to have cohomological dimension 2. An analysis is given of the work of Koch where he gives a weaker criterion for a pro-p-group to have have cohomological dimension 2. A connecttion is made with this work of Koch and that of the author which would have been sufficient to prove the fact that GS was of cohomological dimension 2 for certain sets S had it been applied to investigate whether the linking condition was true for certain sets S. It is not known if the cohomological dimension of GS is 2 if S does not satisfy this linking condition.

Published in Pure and Applied Mathematics Journal (Volume 13, Issue 4)
DOI 10.11648/j.pamj.20241304.12
Page(s) 59-65
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Pro-p-group, Cohomology, Galois Group, p-extension, Tame Ramification, Lie Algebra, Mild Group, Mild Pro-p-group, Linking Number

References
[1] R. Lyndon, Cohomology theory of groups with a single defining relation, Annals of Math., Series 2, t. 52, (1950), 650-665.
[2] Jean-Pierre Serre, Structure de certains pro-p- groupes, Séminaire Bourbaki, 252, (1963), 1-11.
[3] A. Brumer, Pseudocompact algebras, profinite groups and class formations, Journal of Algebra, 4, (1966), 442- 470.
[4] J. P. Labute, Algèbres de Lie et pro-p-groupes définis par une seule relation, Invent. Math. 4,(1967), 142-158.
[5] H. V. Waldinger, On extending Witt’s formula, J. Algebra 5, 41-58 (1967).
[6] D. Gildenhuys, On pro-p-groups with a single defining relation, Invent. Math. 5 (1968), 357-366.
[7] J. P. Labute, On the descending central series of groups with a single defining relation, J. Algebra 14, 16-23 (1970).
[8] H. Koch, Galois Theory of p-extensions. Springer-Verlag, Berlin, 0222, xiv+190pp.
[9] H. Koch, Über pro-p-gruppen der kohomologischen dimension 2, Math. Nachr. 78 (1977), 285-289.
[10] D. Anick, Non-commutative algebras and their Hilbert series, J. Algebra 78 (1982), 120-140.
[11] J. Labute, The Determination of the Lie algebra associated to the central series of a group, Trans. Amer. Math. Soc. 288 (1985), 51-57.
[12] D. Anick, Inert sets and the Lie algebra associated to a group, J. Algebra 111 (1987), 154-165.
[13] J. Labute, Mild pro-p-groups and Galois groups of p- extensions of ℚ, J. Reine Angew. Math. 596 (2006), 115- 130.
[14] A. Schmidt, Über pro-p-fundamentalgruppen markierte arithmetischer curven, J. Reine Angew. Math., 640 (2010), 203-235.
[15] P. Forré,Strongly free sequences and pro-p-groups of cohomological dimension 2, J. Reine Angew. Math. 658,(2011).
[16] J. Labute, J. Minác, Mild pro-2-groups and 2-extensions of ℚ with restricted ramification, J. Alg, 332 (2011), 136- 158.
[17] J. Gärtner, Higher Massey products in the cohomology of mild pro-p-groups, J. Alg., 422 (2015)788-822.
[18] J. Minác, F. Passini, C. Quadrelli, D.T. Nguy�EA;ñ, Mild pro-p-groups and the Koszulity Conjectures, Expositiones Mathematicae 40.
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    Labute, J. (2024). The Genesis of a Theorem in the Galois Theory of p-Extensions of ℚ with Restricted Tame Ramification. Pure and Applied Mathematics Journal, 13(4), 59-65. https://doi.org/10.11648/j.pamj.20241304.12

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    Labute, J. The Genesis of a Theorem in the Galois Theory of p-Extensions of ℚ with Restricted Tame Ramification. Pure Appl. Math. J. 2024, 13(4), 59-65. doi: 10.11648/j.pamj.20241304.12

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    AMA Style

    Labute J. The Genesis of a Theorem in the Galois Theory of p-Extensions of ℚ with Restricted Tame Ramification. Pure Appl Math J. 2024;13(4):59-65. doi: 10.11648/j.pamj.20241304.12

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  • @article{10.11648/j.pamj.20241304.12,
      author = {John Labute},
      title = {The Genesis of a Theorem in the Galois Theory of p-Extensions of ℚ with Restricted Tame Ramification},
      journal = {Pure and Applied Mathematics Journal},
      volume = {13},
      number = {4},
      pages = {59-65},
      doi = {10.11648/j.pamj.20241304.12},
      url = {https://doi.org/10.11648/j.pamj.20241304.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20241304.12},
      abstract = {This article traces the genesis of a theorem that gives for the first time examples of the Galois group GS of the maximal p-extension of ℚ, unramified outside a finite set of primes not containing an odd p, that are of cohomlogical dimension 2 if the primes in S satisfy a certain linking condition. Because the ramification is tame the pro-p-group GS has all of its derived factors finite which is a strong finitenesss condition on GS. The paper starts with a question of Serre on one relator pro-p-groups and then a detour to discrete groups where the notion of strong freeness for a sequence of homogeneous Lie elements is given and a criterion for strong freeness is established. These notions are then carried over to pro-p-groups where the linking condtion on the primes of S is translated into a cohomological criterion for a pro-p-group to have cohomological dimension 2. An analysis is given of the work of Koch where he gives a weaker criterion for a pro-p-group to have have cohomological dimension 2. A connecttion is made with this work of Koch and that of the author which would have been sufficient to prove the fact that GS was of cohomological dimension 2 for certain sets S had it been applied to investigate whether the linking condition was true for certain sets S. It is not known if the cohomological dimension of GS is 2 if S does not satisfy this linking condition.},
     year = {2024}
    }
    

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    AB  - This article traces the genesis of a theorem that gives for the first time examples of the Galois group GS of the maximal p-extension of ℚ, unramified outside a finite set of primes not containing an odd p, that are of cohomlogical dimension 2 if the primes in S satisfy a certain linking condition. Because the ramification is tame the pro-p-group GS has all of its derived factors finite which is a strong finitenesss condition on GS. The paper starts with a question of Serre on one relator pro-p-groups and then a detour to discrete groups where the notion of strong freeness for a sequence of homogeneous Lie elements is given and a criterion for strong freeness is established. These notions are then carried over to pro-p-groups where the linking condtion on the primes of S is translated into a cohomological criterion for a pro-p-group to have cohomological dimension 2. An analysis is given of the work of Koch where he gives a weaker criterion for a pro-p-group to have have cohomological dimension 2. A connecttion is made with this work of Koch and that of the author which would have been sufficient to prove the fact that GS was of cohomological dimension 2 for certain sets S had it been applied to investigate whether the linking condition was true for certain sets S. It is not known if the cohomological dimension of GS is 2 if S does not satisfy this linking condition.
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