An explicit upper bound for the least prime ideal in the Chebotarev density theorem
Annales de l'Institut Fourier, to appear, 48 p.

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.

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.

Received : 2017-06-23
Revised : 2018-02-22
Accepted : 2018-06-13
Classification:  11R44,  11R42,  11M41,  11R45
Keywords: The Chebotarev density theorem, Dedekind zeta functions, the Deuring–Heilbronn phenomenon
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     author = {Ahn, Jeoung-Hwan and Kwon, Soun-Hi},
     title = {An explicit upper bound for the least prime ideal in the Chebotarev density theorem},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
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, to appear, 48 p.

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