Gauss' Lemma and repeated factors I am reading a proof showing that $x^5+y^7 +z^{11}$ is irreducible in $\mathbb{C}[x,y,z]$. We use the natural isomorphism to show that $\mathbb{C}[x,y,z] \cong \mathbb{C}[y,z][x]$. 
We want to use Eisenstein's criterion to show that there exists $p(x) \in \mathbb{C}[y,z]$ such that $p \mid y^7 + z^{11}$ and $p^2 \nmid y^7 + z^{11}$. 
It makes sense that we want to do this. However, the proof goes on to say that it suffices to find such $p(x) \in k(z)[y]$ where $k(z)$, I'm assuming, is the field of fractions of polynomials in $z$ over $k$. 
Since $y^7 + z^{11}$ is a non-unit over a UFD $k(z)[y]$ it is divisible by some irreducible $p(x) \in k(z)[y]$. 
Furthermore, we can show that $y^7 + z^{11}$ has no repeated factors in $k(z)[y]$. 
I think it is necessary to work in $k(z)[y]$ for the $gcd$ computation giving us that there are no repeated factors. 
However, how does having no repeated factors in $k(z)[y]$ imply that there are no repeated factors in $k[z,y]$? 
My notes say this is by Gauss' lemma. However, Gauss' lemma only tells me that if $R$ is UFD then $R[x]$ is UFD. 

Hint: If $p(y,z)$ is a nonconstant factor of $y^7+z^{11}$ in $k[y,z]$, show that the degree of $y$ in $p(y,z)$ is positive.  This will show that $p(y,z)$ is nonconstant in $k(z)[y]$.  Hence, if $p(y,z)$ is a repeated nonconstant factor of $y^7+z^{11}$ in $k[y,z]$, then it is a repeated nonconstant factor of $y^7+z^{11}$ in $k(z)[y]$.
