## Abstract

Let K be a Hilbertian presented field with elimination theory of characteristic

≠ 2, let Ksymm be the compositum of all symmetric extensions of K, and let Ksymm,ins be the maximal purely inseparable extension of Ksymm. Then, Th(Ksymm,ins) is a primitive recursive theory. Moreover, the set of finite groups that can be realized as Galois groups over K in Ksymm as well as the set of finite groups that occur as Galois groups over Ksymm are primitive recursive subsets of the set of all finite groups. Finally, if K is countable, then Gal(Ksymm/K) ∼= Gal(Qsymm/Q).

≠ 2, let Ksymm be the compositum of all symmetric extensions of K, and let Ksymm,ins be the maximal purely inseparable extension of Ksymm. Then, Th(Ksymm,ins) is a primitive recursive theory. Moreover, the set of finite groups that can be realized as Galois groups over K in Ksymm as well as the set of finite groups that occur as Galois groups over Ksymm are primitive recursive subsets of the set of all finite groups. Finally, if K is countable, then Gal(Ksymm/K) ∼= Gal(Qsymm/Q).

Original language | English |
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Pages (from-to) | 139-161 |

Number of pages | 23 |

Journal | Münster J. Math. |

Volume | 12 |

Issue number | 1 |

DOIs | |

State | Published - 2019 |