A negative result on algebraic specifications of the meadow of rational numbers

Abstract

${\mathbb{Q}}_0$ - the involutive meadow of the rational numbers - is the field of the rational numbers where the multiplicative inverse operation is made total by imposing $0^{-1}=0$. In this note, we prove that ${\mathbb{Q}}_0$ cannot be specified by the usual axioms for meadows augmented by a finite set of axioms of the form $(1+\cdots +1+x^2)\cdot (1+\cdots +1+x^2{)}^{-1}=1$.

5 pages, 2 tables