# prime subfield

The prime subfield^{} of a field $F$ is the intersection of all subfields^{} of $F$, or equivalently the smallest subfield of $F$. It can also be constructed by taking the quotient field of the additive subgroup^{} of $F$ generated by the multiplicative identity^{} $1$.

If $F$ has characteristic^{} $p$ where $p>0$ is a prime, then the prime subfield of $F$ is isomorphic^{} to the field $\mathbb{Z}/p\mathbb{Z}$ of integers mod $p$. When $F$ has characteristic zero, the prime subfield of $F$ is isomorphic to the field $\mathbb{Q}$ of rational numbers.

Title | prime subfield |
---|---|

Canonical name | PrimeSubfield |

Date of creation | 2013-03-22 12:37:47 |

Last modified on | 2013-03-22 12:37:47 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 4 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 12E99 |