I stumbled across a statement without proof that I am having a hard time understanding or providing myself with intuition for. Any help would be most appreciated.
The statement reads:
Since $\xi$ is in $k(\eta)$, this means that there are polynomials $v$, $u$, with coefficients in $k$, and without common factors of positive degree, such that $v(\eta)\not=0$ and $\xi = u(\eta)/v(\eta)$