= Nonfinite generation of elliptic-curve divisor class groups
For an <elliptic curve> over an <algebraically closed field>, $\operatorname{Pic}^0(E)\cong E(k)$ via $P\mapsto[P-O]$. This group is infinite and divisible by any prime $\ell$ different from the characteristic, since $[\ell]:E\to E$ has nonzero derivative and is a surjective morphism of projective curves. A <finitely generated abelian group> divisible by $\ell$ has zero free rank and is finite. Hence $E(k)$, and therefore $\operatorname{Cl}(E)$, cannot be finitely generated. This proof applies to countable algebraically closed fields as well.
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