The Conjugates of Algebraic Schemes

Abstract

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 $ϕ^{\sharp}(k(Y))$ is a subfield of $k(X)$ by $ϕ$. The main theorem of this paper says that $k(X) /ϕ^{\sharp}(k(Y)) $ is a Galois extension and the Galois group $Gal(k(X)/ϕ^{\sharp}(k(Y))) $ is isomorphic to the group of $k-$automorphisms of $X$ over $Y$.