Let $E: y^2 = f(x)$ be plane curve defined over a field of characteristic $0$ where $f \in K[x]$ is a cubic polynomial. In order for $E$ to be an elliptic curve, it must be smooth which means $(\frac{\partial g}{\partial x}(P), \frac{\partial g}{\partial y}(P)) \neq (0,0)$ for all points $P \in E$ where $g = y^2 - f(x)$.
Question Is it true that $E$ is smooth if $f$ is a separable polynomial?
I think that we may can use the fact that $f$ is separable if and only if $\gcd(f,f')=1$. But I don't know if it is possible to establish a connection to the definition of smoothness.
Could you please help me with that problem please?
The discriminant of a polynomial of positive degree is zero if and only if the polynomial has a multiple root. If $f$ is separable over a field of characteristic zero, this is not the case. Hence the discriminant is nonzero and $E$ is smooth.