Let $X$ and $Y$ be affine algebraic groups over an algebraically closed field $k$ of characteristic $0$, and $\phi: X \to Y$ be a group homomorphism, which is also a morphism of varieties. I would like to prove that $\phi$ is an isomorphism.
I know that in characteristic $p$ there is Frobenius map $Fr: \mathbb G_a \to \mathbb G_a, x \mapsto x^p$, which is bijective but not an isomorphism. It corresponds to field extension $k(x^p) \subset k(x)$, which is purely transcendental. I suppose that to start I should prove that, as all field extensions in characteristic 0 are separable, the field extension corresponding to $\phi$ is trivial.
After this, one could use the fact that a bijective birational morphism $\phi$ between irreducible affine varieties $X, Y$, such that $Y$ is normal, is an isomorphism.