[FOM] Definable sets of primes

[Timothy Y. Chow]

Joe Shipman wrote:
> But there is another famous condition on a set of primes that forces it
> to have a density: by Frobenius and Chebotarev, the set of primes
> modulo which a given polynomial has a root has a density. Call such a
> set of primes "polynomially definable".
>
> By quadratic reciprocity, one can show that sets of primes definable by
> QUADRATIC polynomials are additively definable. But is this true for
> all polynomials? In other words, are the classes of additively
> definable and polynomially definable sets of primes the same?

If I understand you correctly, the following example from a paper by 
Robert Langlands ("Representation theory: Its rise and role in number 
theory," Proceedings of the Gibbs Symposium, Yale University,
May 15-17, 1989) is relevant:

    x^5 + 10 x^3 - 10 x^2 + 35 x - 18.

Quoting Langlands: "It is irreducible modulo p for p = 7, 13, 19, 29, 
43, 47, 59, ... and factors into linear factors modulo p for p = 2063, 
2213, 2953, 3631, ... .  These lists can be continued indefinitely, but it 
is doubtful that even the most perspicacious and experienced mathematician 
would detect any regularity.  It is none the less there."

In particular, if you're hoping that the primes p for which that 
polynomial has a linear factor modulo p can be described in terms of 
congruences, you're out of luck.  The way the polynomial factors is 
related to the decomposition law for primes in the corresponding number 
field.  Class field theory gives you (more or less) what you are looking 
for in the case where the Galois group is abelian, but in general, the 
conjectural picture involves automorphic representations of Lie groups.

The particular equation above has an icosahedral Galois group and was 
studied by Joe Buhler in his thesis.

Tim