# biquadratic extension

A biquadratic extension of a field $F$ is a Galois extension^{} $K$ of $F$ such that $\mathrm{Gal}(K/F)$ is isomorphic^{} to the Klein 4-group. It receives its name from the fact that any such $K$ is the compositum of two distinct quadratic extensions of $F$. The name can be somewhat misleading, however, since biquadratic extensions of $F$ have exactly three distinct subfields^{} that are quadratic extensions of $F$. This is easily seen to be true by the fact that the Klein 4-group has exactly three distinct subgroups^{} of order (http://planetmath.org/OrderGroup) 2.

Note that, if $\alpha ,\beta \in F$, then $F(\sqrt{\alpha},\sqrt{\beta})$ is a biquadratic extension of $F$ if and only if none of $\alpha $, $\beta $, and $\alpha \beta $ are squares in $F$.

Title | biquadratic extension |
---|---|

Canonical name | BiquadraticExtension |

Date of creation | 2013-03-22 15:56:21 |

Last modified on | 2013-03-22 15:56:21 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 7 |

Author | Wkbj79 (1863) |

Entry type | Definition |

Classification | msc 11R16 |

Related topic | BiquadraticField |

Related topic | BiquadraticEquation2 |