  We give a new elementary proof of the fact that the value of the least
$k^{th}$ power non-residue in an arithmetic progression $\{bn+c\}_{n=0,1...}$,
over a prime field $\F_p$, is bounded by $7/\sqrt{5} \cdot b \cdot \sqrt{p/k} +
4b + c$. Our proof is inspired by the so called \emph{Stepanov method}, which
involves bounding the size of the solution set of a system of equations by
constructing a non-zero low degree auxiliary polynomial that vanishes with high
multiplicity on the solution set. The proof uses basic algebra and number
theory along with a determinant identity that generalizes both the Sylvester
and the Vandermonde determinant.
