  Proofs that an arbitrary field has a separable closure are necessarily
non-constructive, and separable closures are unique only up to non-canonical
isomorphism. This means that the absolute Galois group of a field is defined
only up to inner automorphism. Here we construct a profinite algebraic group
which is an inner form of the absolute Galois group. Our construction uses no
form of the axiom of choice, and the group is defined up to canonical
isomorphism. We also show that the Frobenius associated with a prime of a
number field unramified in an extension, which is classically defined only up
to conjugation, has a uniquely-defined analogue in terms of our group. We give
a construction of the category of Artin motives with coefficients in an
arbitrary field, and we give an interpretation of our absolute Galois group in
terms of this category.
