The Least Nonsplit Prime in Galois Extensions of Q

✍ Scribed by Jeffrey D Vaaler; José Felipe Voloch

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 141 KB
• Volume: 85
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2000.2551

✦ Synopsis

Let k be a Galois extension of Q with [k : Q]=d 2. The purpose of this paper is to give an upper bound for the least prime which does not split completely in k in terms of the degree d and the discriminant 2 k . Our estimate improves on the bound given by Lagarias et al. [3]. We note, however, that with the assumption of the generalized Riemann hypothesis much stronger bounds have been obtained by Murty [7]. In fact, the analytic method employed in [7] can be used to produce an unconditional bound of the same general type as ours. The case of an abelian extension was considered earlier by Bach and Sorensen [1] and Oesterle [8].

Our method is essentially elementary. It is based on an application of the product formula to the binomial coefficient ( : N ), where : is an irrational algebraic integer in k and Trace kÂQ (:)=0. A similar idea has been used in [11] to give a lower bound on the number of primes that do split completely in k. At one point in our argument we appeal to the prime number theorem with an error term in which all constants are given explicitly. Thus for each d we obtain a bound on the least prime which does not split completely provided |2 k | is large compared with d. In the special case k=Q(-p) a somewhat simpler argument can be used which avoids the prime number theorem and leads to a result which is valid for all discriminants. The simpler argument differs insignificantly from that used by Gauss in the course of his first proof of quadratic reciprocity [2, art. 129]. Theorem 1. If exp(max[105, 25(log d ) 2 ]) 8 |2 k | 1Â2(d&1) , (1.1