Consider a nonzero homogenuous degree three polynomial $P\in k[X,Y,Z]$ in $3$ variables with coefficients in a field $k$ of characteristic $\neq 2, 3$.
- How can one check whether $P$ is irreducible? Is there an analogue for cubics of the discriminant of a quadratic form that decides this question for conics?
- In the first paragraph here it is claimed that an irreducible cubic either has an inflection point or a singular point, and that this follows from Bezout's theorem. Could someone provide details?
I'm starting to learn about elliptic curves : these are some elementary questions I couldn't find answers to online. Regarding the first point, all I have found is this fact which implies that the equation of an elliptic curve in Weierstrass form is irreducible as a polynomial $\in k[X,Y]$. I'm not quite sure whether this fact is answers my first question. Certainly irreducibility of the Weierstrass normal form should imply irreducibility of the homogeneous cubic $P$.
Reducing the equation of an elliptic curve to Weierstrass form requires as far as I understand, sending the flex point and its tangent line to the line at infinity, thus the existence of flex points is important.