$a(x)$, $b(x) \in \mathbb{C}(x)$ and $b(x)^2 = a(x)^3 + 1$ implies $a(x)$, $b(x)$ constant?

Question

If $a(x)$, $b(x) \in \mathbb{C}(x)$ and $b(x)^2 = a(x)^3 + 1$, then does it necessarily follow that $a(x)$ and $b(x)$ are constant?

Edit. To clarify, $\mathbb{C}(x)$ is the field of rational functions in $x$ whose coefficients are in $\mathbb{C}$.

Answer 1

The curve $C$ given by $y^2=x^3+1$ is an affine open subset of the elliptic curve $y^2z=x^3+z^3$.
An elliptic curve has genus one and is not rational, so $C$ is not rational either and thus cannot be parametrized by rational functions $a(x), b(x)$.