  Abhyankar proved that every field of finite transcendence degree over
$\mathbb{Q}$ or over a finite field is a homomorphic image of a subring of the
ring of polynomials $\mathbb{Z}[T_1, \dots, T_n]$ (for some $n$ depending on
the field). We conjecture that his result can not be substantially strengthened
and show that our conjecture implies a well-known conjecture on the additive
idempotence of semifields that are finitely generated as semirings.
