The Conjugates of Algebraic Schemes

Original · EN

Fixed an algebraic scheme Y. We suggest a definition for the conjugate of an algebraic scheme X over Y in an evident manner; then X is said to be Galois closed over Y if X has a unique conjugate over Y. Now let X and Y both be integral and let X be Galois closed over Y by a surjective morphism ϕ of finite type. Then ϕ(k(Y)) is a subfield of k(X) by ϕ. The main theorem of this paper says that k(X) /ϕ(k(Y)) is a Galois extension and the Galois group Gal(k(X)/ϕ(k(Y))) is isomorphic to the group of k-automorphisms of X over Y.