Rationality of a smooth projective genus-zero curve ($g(C)=0⟹C\cong {\mathbb{P}}^1$)

Over an algebraically closed field, choose a point $P$ on the smooth projective curve. The Riemann-Roch theorem gives $\dim L(P)=2$: the complementary canonical divisor term has negative degree. A nonconstant function in $L(P)$ has exactly one simple pole, so it defines a degree-one morphism of algebraic curves to the projective line. A degree-one map between smooth projective curves is an isomorphism. The existence of a rational point is essential over a field which is not algebraically closed.