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Proofs that an arbitrary field has a separable closure are necessarilynon-constructive, and separable closures are unique only up to non-canonicalisomorphism. This means that the absolute Galois group of a field is definedonly up to inner automorphism. Here we construct a profinite algebraic groupwhich is an inner form of the absolute Galois group. Our construction uses noform of the axiom of choice, and the group is defined up to canonicalisomorphism. We also show that the Frobenius associated with a prime of anumber field unramified in an extension, which is classically defined only upto conjugation, has a uniquely-defined analogue in terms of our group. We givea construction of the category of Artin motives with coefficients in anarbitrary field, and we give an interpretation of our absolute Galois group interms of this category.

A choice-free absolute Galois group and Artin motives