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 |
Pro-p-group, Cohomology, Galois Group, p-extension, Tame Ramification, Lie Algebra, Mild Group, Mild Pro-p-group, Linking Number
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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
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
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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Download@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}
}
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 -
Department of Mathematics and Statistics, McGill University Montreal, Quebec, Canada
