Suppose . Does the map defined by leave something fixed?
First of all, is the set with the elements and added in? It is an autmomorphism of E?
leaves fixed. If F is a field and if and is algebraic over F with , the map defined by
is an isomorphism if and only if and are conjugate over F where . In your example can be thought of as and it is an automorphism in E.
can be thought of as an extension field (actually it is an algebraic extension) of a field adjoining to F the elements and that are not elements of .
can also be thought of as a splitting field of .