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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.