Clarification on "polynomial over a field" I found this text from Hoffman & Kunze book (section 7.5):

Suppose $f$ and $g$ are polynomials over $F$, a subfield of the complex numbers. We may also regard $f$ and $g$ as polynomials with complex coefficients. The statement that $f$ and $g$ are relatively prime as polynomials over $F$ is equivalent to the statement that $f$ and $g$ are relatively prime as polynomials over the field of complex numbers. We leave the proof of this as an exercise.

So, different questions arise. 
* i. When we say "a polynomial over a field $F$" that means the coefficients and domain of the polynomial are over the field $F$ right? So for example $f(x) = x+2$ and $g(x) = x+2i$ are polynomials over the complex numbers and we may set $x$ to $5i+2$ 


* ii. What am I supposed to prove here? that if a polynomial is defined over a subfield of $F$ then is also defined over $F$? or the roots are the same? 


Sorry for the mess, I am trying to study on my own but this book is particularly confusing sometimes.
 A: When we say that $f$ is a polynomial over the field $F$, it means that the coefficients of $f$ are elements of $F$, and we write $f(x) \in F[x]$. 
It seems to me that you are confusing the term “polynomials” and “polynomial functions”. A polynomial function is a function $$p:x \mapsto \sum_{k=0}^n a_kx^k,$$ with a given domain. On the other hand, a polynomial is a infinite series of the form $$a_0+a_1x+a_2x^2+...+a_nx^n+...$$ where $x$ doesn’t have a given value (it is called a indeterminate.) You can see a polynomial as a linear combination of the elements of the infinite set $B=\{1,x,x^2,...,x^n,...\}$. In fact, $B$ is the canonical basis of the vector space $F[x]$ of all polynomials with coefficients in $F$.
A: A polynomial is not the polynomial function that it defines, over a finite field, two distinct polynomials may be associated to the same function example $X^p-X$ is zero over the finite field with $p$ elements. 
Here si $F[X]$ and $\mathbb{C}[X]$ are PID, you have to show that  $f=pq$ $deg(P)>1, deg(q)>1$ in $F[X]$, it is equivalent that $f=p'q', deg(p)>1, deg(q)>1$ in $\mathbb{C}[X]$.
