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  "body": "\n## Definition\n\nLet $R$ be an [[integral domain|09S3PFBEPM]].\nA polynomial $f \\in R[x]$ is irreducible over $R$ if it is an [[irreducible element|052SDEPZH7]] of the integral domain $R[x]$.\nThat is, $f$ is nonzero and not a unit, and if $f = g h$ with $g, h \\in R[x]$, then $g$ or $h$ is a unit of $R[x]$.\n\n## Remarks\n\nOver a [[field|0QA8CHH8MG]] $K$, the units of $K[x]$ are the nonzero constants. So $f$ is irreducible if and only if $\\deg f \\ge 1$ and $f$ is not a product of two polynomials of smaller degree.\nThe polynomial $x^2 - 2$ is irreducible over $\\mathbb{Q}$ but not over $\\mathbb{R}$, where $x^2 - 2 = (x - \\sqrt{2})(x + \\sqrt{2})$.\nThe polynomial $x^2 + 1$ is irreducible over $\\mathbb{R}$ but not over $\\mathbb{C}$.\nOver $\\mathbb{Z}$, the polynomial $2x + 2 = 2(x + 1)$ is not irreducible, because $2$ is not a unit of $\\mathbb{Z}[x]$.\n\n## Explanations\n\n### L1 human\n\nIrreducible polynomials are the prime numbers of polynomial rings.\nOver a field, they are the polynomials that you cannot factor into polynomials of smaller degree.\nIrreducibility depends on the field: $x^2 + 1$ factors over $\\mathbb{C}$ but not over $\\mathbb{R}$.\n",
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