Abstract
Let p be a prime. It is a fundamental problem to classify the absolute Galois groups G(F) of fields F containing a primitive pth root of unity xi(p). In this paper we present several constraints on such GF, using restrictions on the cohomology of index p normal subgroups from [LMS]. In section 1 we classify all maximal p-elementary abelian-by-order p quotients of these G(F). In the case p > 2, each such quotient contains a unique closed index p elementary abelian subgroup. This seems to be the first case in which one can completely classify nontrivial quotients of absolute Galois groups by characteristic subgroups of normal subgroups. In section 2 we derive analogues of theorems of Artin-Schreier and Becker for order p elements of certain small quotients of GF. Finally, in section 3 we construct a new family of pro-p-groups which are not absolute Galois groups over any field F.
| Original language | English |
|---|---|
| Pages (from-to) | 175-191 |
| Number of pages | 17 |
| Journal | Journal für die reine und angewandte Mathematik |
| Volume | 2007 |
| Issue number | 613 |
| DOIs | |
| Publication status | Published - Dec 2007 |
Keywords
- extensions
- fields
Cite this
Detecting pro-p-groups that are not absolute Galois groups. / Benson, David John; Lemire, Nicole; Minác, Ján; Swallow, John.
In: Journal für die reine und angewandte Mathematik, Vol. 2007, No. 613, 12.2007, p. 175-191.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Detecting pro-p-groups that are not absolute Galois groups
AU - Benson, David John
AU - Lemire, Nicole
AU - Minác, Ján
AU - Swallow, John
PY - 2007/12
Y1 - 2007/12
N2 - Let p be a prime. It is a fundamental problem to classify the absolute Galois groups G(F) of fields F containing a primitive pth root of unity xi(p). In this paper we present several constraints on such GF, using restrictions on the cohomology of index p normal subgroups from [LMS]. In section 1 we classify all maximal p-elementary abelian-by-order p quotients of these G(F). In the case p > 2, each such quotient contains a unique closed index p elementary abelian subgroup. This seems to be the first case in which one can completely classify nontrivial quotients of absolute Galois groups by characteristic subgroups of normal subgroups. In section 2 we derive analogues of theorems of Artin-Schreier and Becker for order p elements of certain small quotients of GF. Finally, in section 3 we construct a new family of pro-p-groups which are not absolute Galois groups over any field F.
AB - Let p be a prime. It is a fundamental problem to classify the absolute Galois groups G(F) of fields F containing a primitive pth root of unity xi(p). In this paper we present several constraints on such GF, using restrictions on the cohomology of index p normal subgroups from [LMS]. In section 1 we classify all maximal p-elementary abelian-by-order p quotients of these G(F). In the case p > 2, each such quotient contains a unique closed index p elementary abelian subgroup. This seems to be the first case in which one can completely classify nontrivial quotients of absolute Galois groups by characteristic subgroups of normal subgroups. In section 2 we derive analogues of theorems of Artin-Schreier and Becker for order p elements of certain small quotients of GF. Finally, in section 3 we construct a new family of pro-p-groups which are not absolute Galois groups over any field F.
KW - extensions
KW - fields
U2 - 10.1515/CRELLE.2007.096
DO - 10.1515/CRELLE.2007.096
M3 - Article
VL - 2007
SP - 175
EP - 191
JO - Journal für die reine und angewandte Mathematik
JF - Journal für die reine und angewandte Mathematik
SN - 0075-4102
IS - 613
ER -