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  "body": "\n## Statement\n\nLet $r, s \\ge 1$ and let $f_1, \\dots, f_k$ be [[irreducible|00X2A9G9W3]] elements of the [[polynomial ring|0CES91M79A]] $\\Q[t_1, \\dots, t_r, x_1, \\dots, x_s]$, each of positive degree in the variables $x_1, \\dots, x_s$.\nThen there is $a = (a_1, \\dots, a_r) \\in \\Q^r$ such that each $f_i(a, x_1, \\dots, x_s)$ is irreducible in $\\Q[x_1, \\dots, x_s]$.\n\n## Remarks\n\nThe set of good $a$ is large: one may take $a \\in \\Z^r$, and there are infinitely many such $a$.\nClassical proofs expand the roots of $f$ as series in $t$ near infinity and show that rational specialisations giving a factor are rare.\nExample: $x^2 - t$ is irreducible in $\\Q[t, x]$; the value $t = 2$ gives the irreducible $x^2 - 2$, while $t = 4$ gives $(x - 2)(x + 2)$.\nSo irreducibility can fail at some values, and the theorem only says that it survives at many.\nA stronger form preserves the Galois group: if an irreducible $f \\in \\Q(t)[x]$ has Galois group $G$ over $\\Q(t)$, then $f(a, x)$ has Galois group $G$ over $\\Q$ for infinitely many $a \\in \\Q$.\nThe analogue fails over $\\R$: $x^2 - t^2 - 1$ is irreducible in $\\R[t, x]$, but for every $a \\in \\R$ the polynomial $x^2 - a^2 - 1$ has the real roots $\\pm \\sqrt{a^2 + 1}$.\n\n## Explanations\n\n### L1 human\n\nA polynomial in several letters that cannot be factored usually stays unfactorable after you replace some letters by numbers.\nHilbert's theorem says that, over the rationals, you can always choose such numbers.\n\n### L2 human\n\nFields with this property are called [[Hilbertian fields|SFPKB3NC0E]]; number fields are Hilbertian, while finite fields and algebraically closed fields are not.\nThe theorem is the main tool to pass from Galois groups over $\\Q(t)$ to Galois groups over $\\Q$.\nFor example, the generic polynomial with Galois group the full symmetric group $S_n$ over $\\Q(t_1, \\dots, t_n)$ specialises to polynomials over $\\Q$ with Galois group $S_n$.\nThis is the starting point of many attacks on the [[inverse Galois problem|YJS6X8JN7D]].\n",
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