I'm dealing with the plane curve $C=\{(x,y)\in k^2:y^3=x^4+x^3\}$. I want to know if this curve is irreducible, where $k$ is a commutative field.
I know this is equivalent to the ideal $\sqrt{I}$ being prime, where $I=(Y^3-X^4-X^3)\subset k[X,Y]$, but so far I can't tell if $C$ is irreducible or if $I$ is prime.
I'd be most glad if someone tells me if this is true and how to prove. Thank you in advance!
Use Eisenstein to deduce that $y^3-x^3(x+1)$ is irreducible.
As an alternative, note that the polynomial (in the variable $y$) is primitive, of degree $3$ and has no roots. Then invoke the Gauß lemma.