From: Victor Miller Newsgroups: sci.math.research Subject: Re: Explicit Cebotarev density Date: 23 Oct 1996 16:04:03 -0400 pfm@math.ufl.edu (Peter F Mueller) writes: > > Let f be an irreducible polynomial over the integers of degree n. > Let L be its splitting field, and g be an element of the > Galois group. Then Cebotarev's density theorem tells us the > density of the rational primes p such that there is a prime P > of L above p such that g fixes P and induces the Frobenius > automorphism on the corresponding extension of the residue fields. > > I'm wondering if these primes p can be given somehow explicitely > in terms of f. In particular, I'd like to know if these primes > come as a union of primes in certain residue classes m_i mod N. > > The answer of course is yes if f has degree 2 -- it is nothing > else than quadratic reciprocity law then. I actually need the > result for degree 4 (and Galois group S_4), but I guess if that > can be worked out, then also in general. > > Peter M\"uller > > > Your question is treated fairly definitively in the following paper: Jeffrey C. Lagarias, Sets of primes determined by systems of polynomial congruences, Illinois J. Math. 27 (1983), no. 2, 224--239. -- Victor S. Miller | " ... Meanwhile, those of us who can compute can hardly victor@ccr-p.ida.org | be expected to keep writing papers saying 'I can do the CCR, Princeton, NJ | following useless calculation in 2 seconds', and indeed 08540 USA | what editor would publish them?" -- Oliver Atkin