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 - A. Brandis,
   <em> &Uuml;ber die multiplikative Struktur von K&ouml;rpererweiterungen</em>,
   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.