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