### 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

*Journal für die reine und angewandte Mathematik*,

*2007*(613), 175-191. https://doi.org/10.1515/CRELLE.2007.096

**Detecting pro-p-groups that are not absolute Galois groups.** / Benson, David John; Lemire, Nicole; Minác, Ján; Swallow, John.

Research output: Contribution to journal › Article

*Journal für die reine und angewandte Mathematik*, vol. 2007, no. 613, pp. 175-191. https://doi.org/10.1515/CRELLE.2007.096

}

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 -