On p-rationality of number fields. Applications – PARI/GP programs

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, very fast PARI Programs ( § § 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 related to Cohen-Lenstra-Martinet ones and analyse Greenberg's conjecture on the existence of p-rational fields with Galois groups (Z/2Z) t needed for the construction of some Galois representations with open image. We give examples for p = 3, t = 5 and t = 6 ( § § 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 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.

Unlimited Lists of Quadratic Integers of Given Norm Application to Some Arithmetic Properties

“…The following program gives, as t grows from 1 up to B, the Kummer radical M and the integer r obtained from the factorizations of m 1 (t) = t 2 − 1, under the form Mr 2 ; then we put them in a list LM and the F.O.P. algorithm gives the pairs C = core(mt, 1) = [M, r], in the increasing order of the radicals M and removes the duplicate entries: [2,2], [3,1], [5,4], [6,2], [7,3], [10,6], [11,3], [13,180], [14,4], [15,1], [17,8], [19,39], [21,12], [22,42], [23,5], [26,10], [29,1820], [30,2], [31,273], [33,4], [34,6], [35,1], [37,12], [38,6], [39,4], ...…”

Triquadratic p-Rational Fields

“…Remark 1.4. There exists algorithmic methods to determine if a given number field K is p-rational for a given prime p, based on underlying profound algebraic results : see [8] for some useful Pari/Gp algorithms.…”

“…) is not 5-rational, because its quadratic subfield Q( √ 38) is not. (One can test the 5-rationality of these fields using the Pari/Gp algorithm of [8]).…”

Genus theory and ε-conjectures on p-class groups

“…Meanwhile a particular study of the algorithm giving ∆(N ), independently of any density results, should be a crucial step for many questions in number theory. {p=3;f=7*13*19*31*37;V=polsubcyclo(f,p);d=matsize(V);d=component(d,2); for(k=1,d,P=component(V,k);if(nfdisc(P)!=f^(p-1),next);K=bnfinit(P,1); C8=component(K,8);C81=component(C8,1);h=component(C81,1); Cl=component(C81,2);print("p=",p," f=",f," P=",P," Cl=",Cl))} p=3 f=1983163=7*13*19*31*37 Cl=[7,7,7,7] Cl=[253,11]= [23]x [11,11] For p = 2, F N,2 may be real or complex; as we know, the 2-rank is always N − 1 and any exceptional classes give non-trivial 4-ranks. The PARI/GP instruction bnfnarrow allows the 2-structure in the restricted sense:…”