## Abstract

Let F be a finite extension of ℚ, ℚ_{p} or a global field of positive characteristic, and let E/F be a Galois extension. We study the realization fields of finite subgroups G of GL_{n}(E) stable under the natural operation of the Galois group of E/F. Though for suffciently large n and a fixed algebraic number field F every its finite extension E is realizable via adjoining to F the entries of all matrices g ε G for some finite Galois stable subgroup G of GL_{n}(ℂ), there is only a finite number of possible realization field extensions of F if G ⊂ GL_{n}(O_{E}) over the ring O_{E} of integers of E. After an exposition of earlier results we give their refinements for the realization fields E/F. We consider some applications to quadratic lattices, arithmetic algebraic geometry and Galois cohomology of related arithmetic groups.

Original language | English |
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Pages (from-to) | 199-227 |

Number of pages | 29 |

Journal | International Journal of Group Theory |

Volume | 2 |

Issue number | 1 |

Publication status | Published - 2013 |

## Keywords

- Algebraic integers
- Galois groups
- Integral representations
- Realization fields

## ASJC Scopus subject areas

- Algebra and Number Theory