From: Douglas Zare
Subject: Galois group actions without n-cycles
Date: Fri, 22 Oct 1999 17:37:47 -0400
Newsgroups: sci.math
At tea, the question of the maximal ideals of Z[x] came up, and these
are the ideals generated by f(x) and a prime p so that f is irreducible
mod p. The standard question of whether every irreducible polynomial in
Z[x] is irreducible mod p for some p came up, as did the standard
incorrect argument that it would be the case for large enough p and the
standard counterexample, the 8th cyclotomic polynomial:
x^4 + 1
= (x^2-i)(x^2+i)
= (x^2-1)^2 + 2x^2 = (x^2-sqrt(-2)x-1)(x^2+sqrt(-2)x-1)
= (x^2+1)^2 - 2x^2 = (x^2-sqrt(2)x-1)(x^2+sqrt(2)x-1)
The first factorization works for primes congruent to 1 or 5 mod 8, the
second when the prime is 1 or 3 mod 8, and the third when the prime is 1
or 7 mod 8. The Galois group action is U(8) = Z/2+Z/2 acting
transitively on the 4 roots. So, as the Cebotarev density theorem
predicts, with density 1/4 a prime will be 1 mod 8 and split completely
and with density 3/4 a prime will be 3, 5, or 7 mod 8 and will split
into two factors.
My question: How can one get other transitive Galois group actions
without n-cycles? These will correspond to polynomials which are
irreducible in Z[x] but factor mod p for every prime p. Perhaps this
should be revised to ask for primitive group actions, so one cannot just
use compositions of previous examples. Is it reasonable to expect that
it is possible to get not just every group but every transitive group
action over Q?
Douglas Zare