definition: A field $\displaystyle E$ is anextension field of a field$\displaystyle F$ if $\displaystyle F \leq E$. (correct?)

Kronecker's theorem: Let $\displaystyle F$ be a field and let $\displaystyle f(x)$ be a non-constant polynomial in $\displaystyle F[x]$. Then there exists an extension field $\displaystyle E$ of $\displaystyle F$ and an $\displaystyle \alpha \in E$ such that $\displaystyle f(\alpha)=0$.

PROOF: (this is where i have a confusion) $\displaystyle f(x)$ has a factorization in $\displaystyle F[x]$ into polynomials that are irreducible over $\displaystyle F[x]$. Let $\displaystyle p(x)$ 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:

$\displaystyle E=F[x]/(p(x))$ is an extension field of $\displaystyle F$.

now how is that??... the elements of $\displaystyle F$ evenlookdifferent that the elements of $\displaystyle E$. i mean to say thatnoelement of $\displaystyle F$ is in $\displaystyle E$.

I understand that $\displaystyle E$ has a sub-field which is isomorphic to $\displaystyle F$ but according to the definition of an extension field how do we regard $\displaystyle E$ as an extension of $\displaystyle F$??

somebody please comment.

(phewf!)