On cyclic cubic fields

Ennola, Veikko; Turunen, Reino

doi:10.1090/s0025-5718-1985-0804947-3

Cited by 3 publications

(2 citation statements)

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“…Note that, due to Galois action, the integers v p ( # T K ) are even if p ≡ 2 (mod 3) and arbitrary if not (same remark for v p ( # Cℓ K ) and [7] with conductor f K ≤ Bf (see the formulas 2.11 giving the corresponding polynomials defining K), and processes as for the quadratic case. We give first the case p = 3 to see the influence of genera theory; we compute the successive maxima of v p ( # T K ) (in vptor) with the corresponding f K and the polynomial defining the field of conductor f K .…”

Section: Numerical Investigations For Cyclic Cubic Fieldsmentioning

confidence: 88%

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Heuristics in direction of a p-adic Brauer--Siegel theorem

Gras

2018

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Let p be a fixed prime number. Let K be a totally real number field of discriminant D K and let T K be the torsion group of the Galois group of the maximal abelian p-ramified pro-p-extension of K (under Leopoldt's conjecture). We conjecture the existence of a constant Cp > 0 such that log( # T K ) ≤ Cp • log( √ D K ) when K varies in some specified families (e.g., fields of fixed degree). In some sense, we suggest the existence of a p-adic analogue, of the classical Brauer-Siegel Theorem, wearing here on the valuation of the residue at s = 1 (essentially equal to # T K ) of the p-adic ζ-function ζp(s) of K. We shall use a different definition that of Washington, given in the 1980's, and approach this question via the arithmetical study of T K since p-adic analysis seems to fail because of possible abundant "Siegel zeros" of ζp(s), contrary to the classical framework. We give extensive numerical verifications for quadratic and cubic fields (cyclic or not) and publish the PARI/GP programs directly usable by the reader for numerical improvements. Such a conjecture (if exact) reinforces our conjecture that any fixed number field K is p-rational (i.e., T K = 1) for all p ≫ 0. Abelian p-ramification -Main definitions and notationsLet K be a totally real number field of degree d, and let p ≥ 2 be a prime number fulfilling the Leopoldt conjecture in K. We denote by Cℓ K the p-class group of K (ordinary sense) and by E K the group of p-principal global units ε ≡ 1 (mod p|p p) of K.Let's recall from [9,12] the diagram of the so called abelian p-ramification theory, in which K c = KQ c is the cyclotomic Z p -extension of K (as compositum with that of Q), H K the p-Hilbert class field and H pr K the maximal abelian p-ramified (i.e., unramified outside p) pro-p-extension of K.is the group of p-principal units of the completion K p of K at p | p, where p is the maximal ideal of the ring of integers of K p .For any field k, let µ k be the group of roots of unity of k of p-power order. Then put W K := tor Zp U K = p|p µ Kp and W K := W K /µ K , where µ K = {1} or {±1}.

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Section: Numerical Investigations For Cyclic Cubic Fieldsmentioning

confidence: 88%

“…real (totally real cubic fields), of cyclic cubic fields of conductor f , described by the polynomials (see, e.g., [7]):…”

Section: 3mentioning

confidence: 99%