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

John Labute

doi:doi:10.11648/j.pamj.20241304.12

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

John Labute^*

Published in Pure and Applied Mathematics Journal (Volume 13, Issue 4)

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 G_S 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 G_S has all of its derived factors finite which is a strong finitenesss condition on G_S. 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 G_S 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 G_S 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

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[9]	H. Koch, Über pro-p-gruppen der kohomologischen dimension 2, Math. Nachr. 78 (1977), 285-289. https://doi.org/10.1002/mana.19770780125
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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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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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TY  - JOUR
T1  - The Genesis of a Theorem in the Galois Theory of p-Extensions of ℚ with Restricted Tame Ramification
AU  - John Labute
Y1  - 2024/08/26
PY  - 2024
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T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
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PB  - Science Publishing Group
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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.
VL  - 13
IS  - 4
ER  -

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Author Information

John Labute

Department of Mathematics and Statistics, McGill University Montreal, Quebec, Canada

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http://orcid.org/0009-0006-9971-0468

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APA Style

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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ACS 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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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}
}

Copy | Download

TY  - JOUR
T1  - The Genesis of a Theorem in the Galois Theory of p-Extensions of ℚ with Restricted Tame Ramification
AU  - John Labute
Y1  - 2024/08/26
PY  - 2024
N1  - https://doi.org/10.11648/j.pamj.20241304.12
DO  - 10.11648/j.pamj.20241304.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 59
EP  - 65
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20241304.12
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.
VL  - 13
IS  - 4
ER  -

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