Year: 2019
Type: article
Abstract: Let K be a number field. We prove that its ray class group modulo p 2 (resp. 8 ) if p > 2 (resp. p = 2 ) characterizes its p -rationality. Then we give two short and very fast PARI Programs (Sections 3.1, 3.2) testing if K (defined by an irreducible monic polynomial) is p -rational or not. For quadratic fields we verify some densities of 3 -rational fields related to Cohen–Lenstra–Martinet ones and analyse Greenberg’s conjecture on the existence of p -rational fields with Galois groups ( ℤ / 2 ℤ ) t needed for the construction of some Galois representations with open image. We give examples for p = 3 , t = 5 and t = 6 (Sections 5.1, 5.2) and illustrate other approaches (Pitoun–Varescon, Barbulescu–Ray). We conclude about the existence of imaginary quadratic fields, p -rational for all p ≥ 2 (Angelakis–Stevenhagen study on the concept of “minimal absolute abelian Galois group”) which may enlighten a conjecture of p -rationality (Hajir–Maire) giving large Iwasawa μ -invariants of some uniform pro- p -groups. All programs (in “verbatim”) can be used by the reader by simply copied and pasted.
DOI: 10.5802/pmb.35
Author: Georges Gras
Institution: Laboratoire de Mathématiques de Besançon
Language: Latvian
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W2992138293