On the Injectivity of the Homomorphisms from the Automorphism Groups of Fields to the Outer Automorphism Groups of the Absolute Galois Groups

Abstract

In the present paper, we discuss the injectivity of the natural homomorphism from the automorphism group of a given field to the outer automorphism group of the associated absolute Galois group. We prove that this natural homomorphism is injective in the case where, for instance, the given field may be embedded into the field of fractions of some Noetherian local domain of mixed characteristic.