Galois Theory

Abstract

Definition 1.1. An extension L of a field k is said to be primary if the largest algebraic separable extension of k in L coincides with k. Proposition 1.2. Let X be a k-scheme. The following statements are equivalent. (a) For every extension K/k, X ⊗k K is irreducible,i.e., geometrically irreducible. (b) For every finite separable extension K/k, X ⊗k K is irreducible. (c) X is irreducible and if x is a generic point, k(x) is a primary extension of k. Proposition 1.3. Let Ω be an algebraically closed field of K and all extensions of K subextensions of ω. N a Galois extension of a field K, E any extension of K and L = N ∩ E. Then the fields E and N are linearly disjoint over L, i.e., E(N) ∼= E ⊗L N . Gal(E(N)/E) ∼= Gal(N/(E ∩ N))