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  "body": "\n## Summary\n\nA perfect field is a field over which every [[irreducible polynomial|03FKX7Z5DC]] has distinct roots.\nEquivalently, it has characteristic $0$, or it has characteristic $p \u003e 0$ and every element is a $p$-th power.\nOver a perfect field every algebraic extension is [[separable|0WCX7NB4N7]], so Galois theory applies without restrictions.\nThe fields $\\Q$, $\\R$, $\\C$, all [[finite fields|0ZXZ82FTX2]] and all algebraically closed fields are perfect.\n\n## Explanations\n\n### L1 human\n\nIn characteristic $p$ the map $x \\mapsto x^p$ is the [[Frobenius endomorphism|0T3WVTD675]], and it is always one-to-one.\nA field of characteristic $p$ is perfect when this map is also onto, so each element has a $p$-th root.\nIn a finite field a one-to-one map from the field to itself is automatically onto, so finite fields are perfect.\nThe field $\\F_p(t)$ of rational functions is not perfect, since $t$ has no $p$-th root in it.\nImperfect fields are where inseparable polynomials such as $x^p - t$ appear.\n",
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