= Countability of integer polynomial rings
{title2=$|\mathbb Z[X_1,\ldots,X_d]|=\aleph_0$}
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.
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