Countability of integer polynomial rings ($|\mathbb{Z}[X_1,\dots ,X_d]|={\aleph }_0$)

For every fixed positive integer $d$, the polynomial ring with integer coefficients in $d$ indeterminates is countably infinite. Polynomials of bounded total degree have only finitely many possible monomials, so they are encoded by a finite Cartesian power of the integers. Taking a countable union of countable sets over degree bounds proves countability; constant polynomials supply an infinite subset. Nonzero one-variable polynomials have finitely many real roots, so the real algebraic numbers form a countable set as well.