Hilbert’s Proof of His Irreducibility Theorem

Hilbert’s irreducibility theorem is a cornerstone that joins areas of analysis and number theory. Both the genesis and genius of its proof involved combining real analysis and combinatorics. We try to expose the motivations that led Hilbert to this synthesis. His famous cube lemma anchored the proof but without the analytical foundation and framework it would have had no purpose. We also assess this lemma as a precursor of Ramsey theory.