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  "body": "\n## Definition\n\nA [[field|0QA8CHH8MG]] $K$ is algebraically closed if every polynomial $f \\in K[x]$ with $\\deg f \\ge 1$ has a [[root|00EHX6FWC0]] in $K$.\n\n## Remarks\n\nEquivalently, every nonzero polynomial over $K$ [[splits|0SHMC00RJV]] over $K$. Also equivalently, the [[irreducible polynomials|00X2A9G9W3]] over $K$ are exactly the polynomials of degree one.\nAnother equivalent condition: $K$ has no [[algebraic extension|09EYMKMSB9]] other than $K$ itself.\nThe field $\\mathbb{C}$ is algebraically closed, by the fundamental theorem of algebra. The field $\\mathbb{R}$ is not, because $x^2 + 1$ has no real root.\nNo finite field is algebraically closed: for a finite field $K$, the polynomial $1 + \\prod_{a \\in K} (x - a)$ has no root in $K$.\n\n## Explanations\n\n### L1 human\n\nIn an algebraically closed field, every polynomial equation of degree at least one has a solution.\nYou never need a bigger field to find roots.\n",
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