Preliminaries on Fields

1 Review on fields We review briefly well-know results in field theory. The notions and results specific to positive characteristic (p-bases, p-independence, etc... ) can be found in Bourbaki [2]. The other unreferenced results come from Chapter III of Lang’s book [38]. We assume a good knowledge of Galois theory. 1.1. Notation. Let A be a field. Then A denotes the (field-theoretic algebraic) closure of A, and A the separable closure of A (i.e., the elements of A which are separably algebraic over A). We denote by G(A) the absolute Galois group of A, i.e., G(A) = Gal(A/A). We often identify G(A) with Aut(A/A). If A and B are subfields of some larger field Ω, we denote by A[B] or B[A] the subring of Ω generated by A and B, and by AB the quotient field of A[B]. ∗partially supported by MRTN-CT-2004-512234 and by ANR-06-BLAN-0183