STNB2025 (38th edition)

Basics on Hilbertian fields

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Alberto Fernandez Boix

Resumen

Hilbert's irreducibility theorem says that if $f\in\mathbb{Q}[T_1,\ldots,T_r,X]$ is an irreducible polynomial, then there are $(a_1,\ldots,a_r)\in\mathbb{Q}^r$ such that $f(a_1,\ldots,a_r,X)\in\mathbb{Q}[x]$ remains irreducible. The goal of this lecture is to formally introduce the so--called \textit{Hilbertian fields}, namely, fields where the above statement is also valid. The main reference for this lecture will be \cite[Chapter 13]{FriedJardenFieldArithmeticbook}.

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