Fields with indecomposable multiplicative groups

We classify all finite fields and all infinite fields of characteristic not equal to 2 whose multiplicative groups are direct-sum indecomposable. For finite fields, we obtain our classification using a direct argument and also as a corollary to Catalan’s Conjecture. Our answer involves both Fermat and Mersenne primes. Turning to infinite fields, we use the classification of indecomposable non-torsion-free abelian groups to prove that any infinite field whose characteristic is not equal to 2 must have a decomposable multiplicative group.