Past exam of the model theory course of the University of Oxford / 2011 / 1 / b / ii / α / Solution

Yes. Zero is the unique absorbing element of the multiplicative first-order structure. It is defined byZero satisfies this universal formula. Conversely, substituting the actual integer zero for $y$ in the semantic verification forces $x=0$. This use of zero in the verification does not insert a constant symbol into the first-order formula.