Suppose $k$ is an algebraically closed field with characteristic 0. Suppose $f(x,y)\in k[x,y]$ is irreducible and viewing $f(x,y)$ as a polynomial over $k[x]$ which is monic in $y$ and of degree>1 in $y$.
We want to prove that the ideal $(f(x,y),f_y(x,y))\neq k[x,y]$. (if it is true)
Is this statement true for general? For example, let $R$ be a domain of dimension$\geq 1$ with char 0 and suppose $f(y)\in R[y]$ is a monic irreducible polynomial of degree >1. Is it true that the ideal $(f(y),f^{\prime}(y))\neq R[y]$ ?
Thanks.