### 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 |
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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