Euclid's Lemma for polynomials I need your help with this:
Let $f$, $p$, and $g$ be a polynomials in $F[x]$, assume that $f$, $g$, and $p$ are nononzero.
If $\gcd(f,p)=1$ and $fg$ is divisible by $p$,
I need to prove that $g$ is divisible by $p$.
Thanks.
 A: Proof $\ \ \ p\mid fg,pg\ \Rightarrow\ p\mid (fg,pg) = (f,p)\ g\ =\ g,\ $ by $\ (f,p) = 1.\quad\ $ QED
Thus if $\,p\,$ is irreducible and $\,p\nmid f\,$ then $\, p\mid fg\Rightarrow p\mid g,\,$ i.e. irreducibles are prime.
This proof of Euclid's Lemma works in any GCD domain, e.g. any domain like $\rm\:F[x]\:$ enjoying a Euclidean algorithm to compute the GCD. See also this answer where I present this version of Euclid's lemma in Bezout, gcd, and ideal form.
A: Polynomials over a field are an Euclidean domain relative to the degree function.  In particular, for every polynomials $a(x)$ and $b(x)$ in $F[x]$, with $b(x)\neq 0$, there exist unique polynomials $q(x),r(x)\in F[x]$ such that
$$ a(x) = q(x)b(x) + r(x),\quad\mbox{$r(x)=0$ or $\mathrm{deg}(r)\lt \mathrm{deg}(b)$}$$
So the Euclidean algorithm also applies to polynomials. In particular, if $q(x) = \gcd(m(x),n(x))$, then there exist polynomial $\alpha(x),\beta(x)$ such that $q(x) = \alpha(x)m(x) + \beta(x)n(x)$.
Now you can use the same proof as the proof of Euclid's Lemma for integers: if $a|bc$ and $\gcd(a,b)=1$, then $a|c$. 
A: If $gcd(f, p) = 1$ then $af + bp = 1$ for some $a, b \in \Bbb{Z}$. But multiplying by $g$ on each side, $g = gaf + gbp$ and therefore $g = a(fg) + bg(p)$.
Let $fg = cp$ for some $c \in \Bbb{Z}$. Then $g = p(ac + bg)$ implying $p|g$. QED.
