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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $f = x^n + a_{n-1} x^{n-1} + \\dots + a_1 x + a_0$ be a [[monic polynomial|0BDX4MBKT4]] with integer coefficients and $a_0 \\ne 0$.\nIf\n$$\\abs{a_{n-1}} \u003e 1 + \\abs{a_{n-2}} + \\dots + \\abs{a_1} + \\abs{a_0},$$\nthen $f$ is [[irreducible|00X2A9G9W3]] in $\\Z[x]$ and in $\\Q[x]$.\n\n## Remarks\n\nProof idea: by [[Rouché's theorem|0D430QR02W]], comparing $f$ with $a_{n-1} x^{n-1}$ on the unit circle, $f$ has exactly $n - 1$ roots in the open unit disc and one root outside it.\nIf $f = g h$ with monic $g, h \\in \\Z[x]$ of positive degree, one factor has all its roots in the open unit disc.\nIts constant term is then a nonzero integer of absolute value less than $1$, which is impossible.\nIrreducibility over $\\Q$ follows by [[Gauss's lemma|0QFV5Y1Z5X]].\nExample: $x^3 + 5x^2 + 2x + 1$ is irreducible, since $5 \u003e 1 + 2 + 1$.\nThe hypothesis $a_0 \\ne 0$ is necessary: $x^2 + 3x = x(x + 3)$ satisfies the inequality $3 \u003e 1 + 0$.\nA second form of the criterion allows equality in the inequality, provided that $f(1) \\ne 0$ and $f(-1) \\ne 0$.\n\n## Explanations\n\n### L1 human\n\nIf the second coefficient of a monic integer polynomial is larger than all the other coefficients together, plus one, the polynomial cannot be factored.\nThe check needs only additions of absolute values.\n\n### L2 human\n\nThe criterion is a typical analytic proof of irreducibility.\nOne locates the complex roots, then uses that the constant term of an integer factor is a nonzero integer.\nIt complements algebraic tests such as [[Eisenstein's criterion|02BPKD7TMD]] and [[reduction modulo a prime|0KWWEXW0RN]].\nFor example, it shows at once that $x^n + 3x^{n-1} + 1$ is irreducible for every $n \\ge 2$.\n",
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