How to show that I is a Prime ideal

Question

I want to show that $ I= (x_2 ^ 2 - x_1 ^2 - x_1 ^3) $ is a prime ideal.

I know that I should show that the quotient of the $K[x_1, x_2]/ I$ is an integral domain. I know that all the powers of the $x_1$ are 1 or 2 in this quotient, but I don't know what should I have to do. Any help would be great thanks.

Answer 1

Consider $x_2^2 - x_1^2 - x_1^3$ as a polynomial in $k[x_1][x_2]$. Since $-x_1^2 - x_1^3 = -x_1^2(x_1 + 1) \in (x_1 + 1)$ and $(x_1 + 1)$ is prime in $k[x_1]$, Eisenstein's criterion says $x_2^2 - x_1^2 - x_1^3$ is irreducible. Thus, the corresponding ideal is prime.