The least non-split prime in a number field

Let K be an S n -field ( n ≤ 5 ) with discriminant d K . Let n K be the least prime which does not split completely in K . We prove unconditionally that n K = O ( log | d K | ) , except for O X exp - c log X log log X fields for some constant c > 0 . We also prove that the exceptional set is opti...

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Bibliographic Details
Published inArchiv der Mathematik Vol. 117; no. 5; pp. 509 - 513
Main Author Kim, Henry H.
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.11.2021
Springer Nature B.V
Subjects
Mathematics
Mathematics and Statistics
Number theory
Secondary 11M41
Artin
Primary 11R42
Least prime in a conjugacy class
Chebotarev density theorem
functions
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Summary:Let K be an S n -field ( n ≤ 5 ) with discriminant d K . Let n K be the least prime which does not split completely in K . We prove unconditionally that n K = O ( log | d K | ) , except for O X exp - c log X log log X fields for some constant c > 0 . We also prove that the exceptional set is optimal, and that for any η > 0 , the S n -fields such that n K ≫ ( log | d K | ) 1 + η are very rare.
ISSN:0003-889X
1420-8938
DOI:10.1007/s00013-021-01650-9