  The multiplicative group of a finite field is well known to be cyclic; in
this note, we determine the finite fields whose multiplicative groups are
direct sum indecomposable. We obtain our classification using a direct argument
and also as a corollary to Catalan's Conjecture. Turning to infinite fields, we
prove that any infinite field whose characteristic is not equal to 2 must have
a decomposable multiplicative group. We conjecture that this is also true for
infinite fields of characteristic 2 and we narrow the class of possible
counter-examples. Finally, using the classification of finite commutative
primary rings with cyclic multiplicative groups, we determine all finite
commutative rings with indecomposable multiplicative groups.
