An explicit upper bound for the least prime ideal in the Chebotarev density theorem
[Une borne explicite pour le plus petit idéal premier dans le théorème de densité de Chebotarev]
Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1411-1458.

Lagarias, Montgomery, et Odlyzko ont démontré qu’il existe une constante absolue effectivement calculable A 1 telle que pour chaque extension finie K de , chaque extension galoisienne finie L de K à groupe de Galois G, et chaque classe de conjugaison C de G, il existe un idéal premier 𝔭 de K qui est nonramifié dans L, pour lequel L/K 𝔭=C, pour lequel N K/ 𝔭 est un nombre premier rationel, et qui satisfait N K/ 𝔭2d L A 1 . Dans cet article nous démontrons sans aucune restriction que N K/ 𝔭d L 12577 si L, en suivant la méthode developpée par Lagarias, Montgomery, et Odlyzko.

Lagarias, Montgomery, and Odlyzko proved that there exists an effectively computable absolute constant A 1 such that for every finite extension K of , every finite Galois extension L of K with Galois group G and every conjugacy class C of G, there exists a prime ideal 𝔭 of K which is unramified in L, for which L/K 𝔭=C, for which N K/ 𝔭 is a rational prime, and which satisfies N K/ 𝔭2d L A 1 . In this paper we show without any restriction that N K/ 𝔭d L 12577 if L, using the approach developed by Lagarias, Montgomery, and Odlyzko.

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DOI : 10.5802/aif.3274
Classification : 11R44, 11R42, 11M41, 11R45
Keywords: The Chebotarev density theorem, Dedekind zeta functions, the Deuring–Heilbronn phenomenon
Mot clés : théorème de densité de Chebotarev, fonction de zêta de Dedekind, phénomène de Deuring–Heilbronn
Ahn, Jeoung-Hwan 1 ; Kwon, Soun-Hi 1

1 Department of Mathematics Education Korea University 02841, Seoul (Korea)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Ahn, Jeoung-Hwan; Kwon, Soun-Hi. An explicit upper bound for the least prime ideal in the Chebotarev density theorem. Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1411-1458. doi : 10.5802/aif.3274. https://aif.centre-mersenne.org/articles/10.5802/aif.3274/

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