We characterize irreducible trinomials x6 + Ax + B with coefficients in a number field K which have Galois group C6, S3 or C3 × S3. This characterization relates these trinomials to the K-rational points on a genus 2 curve. We determine these trinomials explicitly in the case K = ℚ.

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The minimal index of a pure cubic field was shown to assume arbitrarily large values by M. Hall. In this paper we extend this result by showing that every cubefree integer occurs as the minimal index of infinitely many pure cubic fields.