Tchebotarev theorems for function fields

Sara Checcoli (1) , Pierre Dèbes (2)

1 IF - Institut Fourier
2 LPP - Laboratoire Paul Painlevé - UMR 8524

We prove Tchebotarev type theorems for function field extensions over various base fields: number fields, finite fields, p-adic fields, PAC fields, etc. The Tchebotarev conclusion - existence of appropriate cyclic residue extensions - also compares to the Hilbert specialization property. It is more local but holds in more situations and extends to infinite extensions. For a function field extension satisfying the Tchebotarev conclusion, the exponent of the Galois group is bounded by the l.c.m. of the local specialization degrees. Further local-global questions arise for which we provide answers, examples and counter-examples.

Domaines: Théorie des nombres [math.NT]

Sara Checcoli, Pierre Dèbes. Tchebotarev theorems for function fields. Journal of Algebra, 2016, 446, pp.346-372. ⟨10.1016/j.jalgebra.2015.08.020⟩. ⟨hal-01623558⟩