definition: A field is anextension field of a fieldif . (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 evenlookdifferent that the elements of . i mean to say thatnoelement 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.

(phewf!)