  We prove that a commutative parasemifield S is additively idempotent provided
that it is finitely generated as a semiring. Consequently, every proper
commutative semifield T that is finitely generated as a semiring is either
additively constant or additively idempotent. As part of the proof, we use the
classification of finitely generated lattice-ordered groups to prove that a
certain monoid associated to the parasemifield S has a distinguished
geometrical property called prismality.
