Finite Extensions of ${\mathbb{Z}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$

We classify the semifields and division semirings containing the max-plus semifield ${\mathbb{Z}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, which are finitely generated as ${\mathbb{Z}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$-semimodules.