# perfect field

A *perfect field ^{}* is a field $K$ such that every algebraic extension

^{}field $L/K$ is separable

^{}over $K$.

All fields of characteristic 0 are perfect, so in particular the fields $\mathbb{R}$, $\u2102$ and $\mathbb{Q}$ are perfect. If $K$ is a field of characteristic $p$ (with $p$ a prime number^{}), then $K$ is perfect if and only if the Frobenius endomorphism $F$ on $K$, defined by

$$F(x)={x}^{p}\mathit{\hspace{1em}}(x\in K),$$ |

is an automorphism^{} of $K$. Since the Frobenius map is always injective^{}, it is sufficient to check whether $F$ is surjective^{}. In particular, all finite fields^{} are perfect (any injective endomorphism is also surjective). Moreover, any field whose characteristic is nonzero that is algebraic^{} (http://planetmath.org/AlgebraicExtension) over its prime subfield^{} is perfect. Thus, the only fields that are not perfect are those whose characteristic is nonzero and are transcendental over their prime subfield.

Similarly, a ring $R$ of characteristic $p$ is perfect if the endomorphism $x\mapsto {x}^{p}$ of $R$ is an automorphism (i.e., is surjective).

Title | perfect field |
---|---|

Canonical name | PerfectField |

Date of creation | 2013-03-22 13:08:23 |

Last modified on | 2013-03-22 13:08:23 |

Owner | sleske (997) |

Last modified by | sleske (997) |

Numerical id | 11 |

Author | sleske (997) |

Entry type | Definition |

Classification | msc 12F10 |

Related topic | SeparablePolynomial |

Related topic | ExtensionField |

Defines | perfect |

Defines | perfect ring |