Depends on $k$ and $n$.  If $k$ is algebraically closed and of characteristic zero, then the answer is No for $n=2$ by a theorem of Castelnuovo, but Yes for $n=3$ and presumably for all $n \geq 3$.  In positive characteristic things can get much stranger: for each $n \geq 2$ there are inseparable extensions $K/F$ where $F$ is the function field of a variety of general type.  Shioda gave explicit examples where $F$ is the function field of the Fermat hypersurface of degree $q+1$ and $q$ is any power of the characteristic; see Propositions 1 and 3 of

> Shioda, T.: An Example of Unirational Surfaces in Characteristic $p$,
> *Math. Ann.* 211 (1974), 233-236.


The exposition and references at
http://en.wikipedia.org/wiki/Rational_variety may also be of interest.