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• A. Brandis, Über die multiplikative Struktur von Körpererweiterungen, Math. Z. 87 (1965), 71-73

Brandis proved that $L^\times/K^\times$ is not finitely generated whenever $K$ is infinite. The claim is reduced to finite algebraic extensions of global fields, for which there are infinitely many prime ideals in $K$ that do not remain inert in $L$, and this does it.

You're looking for

• A. Brandis, Über die multiplikative Struktur von Körpererweiterungen, Math. Z. 87 (1965), 71-73

Brandis proved that $L^\times/K^\times$ is not finitely generated whenever $K$ is infinite and $L \ne K$ (thanks Pete). The claim is reduced to finite algebraic extensions of global fields, for which there are infinitely many prime ideals in $K$ that do not remain inert in $L$, and this does it.