The least nonsplit prime in Galois extensions of \({\mathbb{Q}}\). (English) Zbl 0963.11066

The purpose of this paper is to give an upper bound for the least rational prime which does not split completely in a finite extension \(k\) of the field of rational numbers \(\mathbb Q\) in terms of the degree \(d =[k:{\mathbb Q}]\) and the discriminant \(\delta _k\) of \(k\). The estimate given here improves on the bound given by J. C. Lagarias, H. L. Montgomery and A. M. Odlyzko [Invent. Math. 54, 271-296 (1979; Zbl 0413.12011)]. The method is based on an application of the product formula to the binomial coefficient \(\alpha \choose N\), where \(\alpha\) is an irrational algebraic integer in \(k\) and \(\text{Trace} _{k/{\mathbb Q}} (\alpha) = 0\). During the argument, the authors use the explicit error term in the prime number theorem obtained by J. B. Rosser and L. Schoenfeld [Math. Comput. 29, 243-269 (1975; Zbl 0295.10036)]. Thus for each \(d\) a bound on the least prime which does not split completely is obtained provided that \(|\delta _k|\) is large compared with \(d\).


11R32 Galois theory
11R45 Density theorems
Full Text: DOI


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