A problem about irreducible polynomials in two indeterminates over a field

Question

I want to know whether it is true that over an arbitrary field $K$, and for any positive integer $m$, does there always exist a prime (or equivalently, irreducible, since the polynomial ring over a field is UFD) polynomial in $K[x, y]$ with degree great than $m$. Moreover, can we choose a prime polynomial with degree exactly $m$?

Answer 1

This is a good application of the general form of Eisenstein's criterion. Namely $x+1$ is prime in $k[x]$ and so $y^n+x+1 \in k[x,y]$ is irreducible for any $n$.