Hilbert's Proof of His Irreducibility Theorem

TL;DR

This paper explores Hilbert's proof of his Irreducibility Theorem, emphasizing the synthesis of real analysis and combinatorics, and highlights the foundational role of Hilbert's Cube Lemma in this context.

Contribution

It provides a detailed analysis of the motivations and analytical foundations behind Hilbert's proof, connecting it to broader mathematical theories.

Findings

Hilbert's Cube Lemma was crucial for the proof.

The proof combines real analysis and combinatorics.

The lemma anticipates aspects of Ramsey Theory.

Abstract

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. Hilbert's famous Cube Lemma supplied fuel for the proof but without the analytical foundation and framework it would have been heating empty air. The lemma is said to presage Ramsey Theory but we note differences in motivation.