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????

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- August 22nd 2009, 03:39 AMynjalgebraic elements and evaluation homomorphism
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???? - August 22nd 2009, 04:17 AMNonCommAlg
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. - August 22nd 2009, 04:36 AMynj
emm...I ignored the fact that is irreducible..