Abstract
We determine the irreducible trinomials [Formula: see text] for integers [Formula: see text] which generate precisely all possible Galois extensions of degree [Formula: see text] over [Formula: see text]. 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 [Formula: see text] or [Formula: see text] mod [Formula: see text] are sums of two rational cubes - thereby, giving the first unconditional result on a classical open problem.
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