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  "body": "\n## Statement\n\nLet $K$ be a [[field|0QA8CHH8MG]] and let $f \\in K[x]$ with $\\deg f \\ge 1$.\nThen there is a [[field extension|0WQ1A8SHTZ]] $L/K$ with $[L : K] \\le \\deg f$ such that $f$ has a [[root|00EHX6FWC0]] in $L$.\n\n## Remarks\n\nThe proof: let $g$ be an irreducible factor of $f$ in $K[x]$. [[Adjoining a root of $g$|0M893X7YY7]] gives the field $L = K[x]/(g)$ of degree $\\deg g$ over $K$.\nThe class of $x$ in $L$ is a root of $g$, and so of $f$.\nThe irreducible factor $g$ exists because $K[x]$ is a [[unique factorisation domain|0D6KH8Y59N]].\nRepeating the step gives a [[splitting field|02NZZ1J0FH]] of $f$.\n\n## Explanations\n\n### L1 human\n\nA polynomial with no root in a field gets a root in a bigger field.\nThe bigger field is built from the polynomial itself, without any reference to the complex numbers.\n",
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