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  "body": "\n## Summary\n\nA field is Hilbertian if irreducible polynomials in several variables stay irreducible after suitable values in the field are substituted for all but one variable.\n[[Hilbert's irreducibility theorem|N1V3YEYHNP]] says that the field $\\Q$ of rational numbers is Hilbertian.\nHilbertian fields are the natural setting for realizing finite groups as [[Galois groups|0XTY86BZKY]] by specialization, an approach to the [[inverse Galois problem|YJS6X8JN7D]].\nIn characteristic $0$, a field is Hilbertian exactly when its affine line is not a thin set in the sense of Serre.\n\n## Explanations\n\n### L1 human\n\nTake a polynomial such as $X^2 - T$, which cannot be factored as a polynomial in $X$ and $T$.\nIf you put a number in place of $T$, the result may factor: $X^2 - 4 = (X - 2)(X + 2)$.\nOver the rational numbers, most choices, such as $T = 2$ or $T = 3$, keep it irreducible.\nA field is Hilbertian if this always happens: for any irreducible polynomial in $X$ and $T$ that involves $X$, some (in fact infinitely many) values of $T$ keep it irreducible.\nThe real numbers are not Hilbertian, since $X^2 - T^2 - 1$ factors for every real value of $T$.\n",
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