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?
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 .