The map $t\to (a(t),b(t),c(t))$, if non-constant, would extend to a map from a lower-genus (compact connected) curve ($\mathbb P^1$) to a higher-genus such, the elliptic curve defined by $a^3+b^3=c^3$. (Maybe open mapping easily shows surjectivity in the complex case, for example.) Impossible, by Riemann-Hurwitz formula.

(With corrections noted by @GH from MO): The map $t\to (a(t),b(t),c(t))$, if at least one of the ratios $a(t)/c(t)$ or $b(t)/c(t)$ is non-constant, would extend to a non-constant map from a lower-genus (compact connected) curve ($\mathbb P^1$) to a higher-genus such, the elliptic curve defined by $a^3+b^3=c^3$. (Maybe open mapping easily shows surjectivity in the complex case, for example.) Impossible, by Riemann-Hurwitz formula.

Also, in the complex case, since elliptic curves are quotients $\mathbb C/L$ by lattices, and $\mathbb C\mathbb P^1$ is simply-connected, a map to the elliptic curve would lift (by homotopy lifting...) to a map to $\mathbb C$. This would contradict Liouville's theorem.