definition: A field is an extension field of a field if . (correct?)
Kronecker's theorem: Let be a field and let be a non-constant polynomial in . Then there exists an extension field of and an such that .
PROOF: (this is where i have a confusion) has a factorization in into polynomials that are irreducible over . Let be an irreducible polynomial in such a factorization. (till here everything is fine).
i am now skipping some details and coming to the point which troubles me:
is an extension field of .
now how is that??... the elements of even look different that the elements of . i mean to say that no element of is in .
I understand that has a sub-field which is isomorphic to but according to the definition of an extension field how do we regard as an extension of ??
somebody please comment.