= Solution
The generator $\eta=f\,dh$ is a nonzero <holomorphic differential form>. Its <canonical divisor> $(\eta)$ is an <effective divisor>, since there are no <poles>. In <genus> one, the <degree of a divisor> formula in part (a) gives
$$
\deg(\eta)=2g-2=0.
$$
An <effective divisor> of degree zero has every coefficient zero. Thus
$$
\boxed{(\eta)=0,\qquad f\,dh\text{ has no zeros anywhere on }C.}
$$
This proves <nonvanishing of holomorphic differentials in genus one>. It concerns the differential itself: a zero or <pole> of the particular coefficient $f$ may be cancelled by the local behavior of $dh$.
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