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