= Solution
The <genus-degree formula> gives
$$
g(C)=p_a(C)=\frac{(3-1)(3-2)}2=1,
$$
because the <nonsingular plane cubic> is already its own <normalization of an algebraic curve>. If it were <birational> to the <projective line>, the two <smooth projective curves> would have isomorphic <function fields>. The uniqueness of the <finite normalization model of a one-variable function field> would then make them isomorphic. But the <projective line> has <genus of a smooth projective curve> zero. Therefore \b[the cubic is not rational]. This obstruction works in the stated positive characteristics as well as in characteristic zero.
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