Cyclic cubic extensions of â„š

We determine the irreducible trinomials X3-aX + b for integers a,b which generate precisely all possible Galois extensions of degree 3 over ℚ. The proof, although involved, is elementary and one can parametrize all these polynomials explicitly. As an accidental by-product of the results, we prove that infinitely many primes congruent to 1 or-1 mod 9 are sums of two rational cubes-thereby, giving the first unconditional result on a classical open problem.