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????
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.