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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]] and let $f \\in K[x]$ be nonzero.\nA splitting field of $f$ over $K$ is a [[field extension|0WQ1A8SHTZ]] $L/K$ with these two properties.\n\n1. The polynomial $f$ [[splits|0SHMC00RJV]] over $L$, as $f = c \\, (x - a_1) \\cdots (x - a_n)$.\n2. The field $L$ is [[generated over $K$|0MA5B744K3]] by the roots: $L = K(a_1, \\dots, a_n)$.\n\n## Remarks\n\nSplitting fields [[exist|01441KDGBY]] and are [[unique up to isomorphism|069A3BC3EP]].\nThe splitting field of $x^2 + 1$ over $\\mathbb{R}$ is $\\mathbb{C}$.\nThe splitting field of $x^3 - 2$ over $\\mathbb{Q}$ is $\\mathbb{Q}(\\sqrt[3]{2}, \\omega)$ with $\\omega = e^{2\\pi i/3}$. Its degree over $\\mathbb{Q}$ is $6$.\nThe splitting field of $x^q - x$ over $\\mathbb{F}_p$, for $q = p^n$, is the [[finite field|08RTDHX1SK]] with $q$ elements.\n\n## Explanations\n\n### L1 human\n\nA splitting field is the smallest field that contains all the roots of a polynomial.\nIt is where the polynomial factors completely.\nFor $x^3 - 2$, the real cube root of $2$ is not enough, because the other two roots are not real.\n",
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