I'm reading a proof of the theorem that $F(a)$ is isomorphic to $ F[x]/\langle p(x) \rangle$ in the book called Contemporary Abstract Algebra by Gallian, and I don't understand some key elements of the proof [I think this book is too confusing in numerous places in the sections on rings and fields, but it was clear and straightforward on groups]. Here are the theorem and the proof:
Here's what I do not understand:
(1) If $\langle p(x) \rangle$ is a maximal ideal in $F[x]$ and $\ker \phi\ne F[x]$, then how does it follow from the First Isomorphism Theorem for Rings and the corollary (below) that $\phi(F[x])$is a subfield of $F(a)$?
(2) When the author writes that
$\phi(F[x])$ contains both $F$ and $a$ and recalling that $F(a)$ is the smallest such field ...
what does he refer to by "the smallest such field"? The smallest what field?
(3) Can't one immediately see the image of $\phi$ is indeed $F(a)$, just by definition? $\phi$ takes all polynomials in $F[x]$ and evaluates them at $a$. I don't understand the author's argument at all.
(4) The author also writes that
But isn't the basis for $F(a)$ one-dimensional, after all?
Sorry for the long message. Would really appreciate your help in understanding this proof. In my opinion the author is skipping too many details. In my university, I'd get reduced marks for unproved statements like "clearly, [...]".