Let F be a subfield of E, and is an algebraic over . Let be the evaluation homomorphism. is the minimal field containing
True or false: ?
One main problem is: is a field????
yes! let be the minimal polynomial of i.e. is monic and it has the smallest degree among all with this property that then and, since
is surjective, we have finally, since is irreducible and is a PID, the ideal is maximal and thus is a field. as a result
the converse is also true: if is a field, then is algebraic over the reason is that if is transcendental over then and obviously is never a field.