An explicit upper bound for the least prime ideal in the Chebotarev density theorem
Annales de l'Institut Fourier, Volume 69 (2019) no. 3, p. 1411-1458

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
Published online : 2019-06-03
DOI : https://doi.org/10.5802/aif.3274
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},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {3},
     year = {2019},
     pages = {1411-1458},
     doi = {10.5802/aif.3274},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2019__69_3_1411_0}
}
An explicit upper bound for the least prime ideal in the Chebotarev density theorem. Annales de l'Institut Fourier, Volume 69 (2019) no. 3, pp. 1411-1458. doi : 10.5802/aif.3274. https://aif.centre-mersenne.org/item/AIF_2019__69_3_1411_0/

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