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  "body": "\n## Definition\n\nLet $K$ be a field.\nThe field $K$ is *perfect* if one of the following holds:\n\n1. $K$ has [[characteristic|0P7CH1CND5]] $0$;\n2. $K$ has characteristic $p \u003e 0$ and the [[Frobenius endomorphism|0GHDDVAH8P]] $x \\mapsto x^p$ of $K$ is surjective, that is, $K^p = K$.\n\n## Remarks\n\nA field $K$ is perfect if and only if every [[irreducible polynomial|00X2A9G9W3]] in $K[x]$ is [[separable|03J8BS7FMV]], if and only if every algebraic extension of $K$ is separable.\n\nExample: a [[finite field|08RTDHX1SK]] $\\F_q$ of characteristic $p$ is perfect.\nCheck: the Frobenius map is an injective ring homomorphism, and an injective map from a finite set to itself is surjective.\n\nNon-example: $K = \\F_p(t)$ is not perfect.\nThe element $t$ is not a $p$-th power.\nIf $(f/g)^p = t$ with polynomials $f, g$, then $f^p = t g^p$.\nThe degree of the left side is divisible by $p$, while the degree of the right side is not.\nAccordingly, $x^p - t$ is irreducible over $K$ but not separable.\n",
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