# norm

Let $K/F$ be a Galois extension^{}, and let $x\in K$. The norm ${\mathrm{N}}_{F}^{K}(x)$ of $x$ is defined to be the product of all the elements of the orbit of $x$ under the group action^{} of the Galois group^{} $\mathrm{Gal}(K/F)$ on $K$; taken with multiplicities if $K/F$ is a finite extension^{}.

In the case where $K/F$ is a finite extension, the norm of $x$ can be defined to be the determinant^{} of the linear transformation $[x]:K\to K$ given by $[x](k):=xk$, where $K$ is regarded as a vector space^{} over $F$. This definition does not require that $K/F$ be Galois, or even that $K$ be a field—for instance, it remains valid when $K$ is a division ring (although $F$ does have to be a field, in order for determinant to be defined). Of course, for finite Galois extensions $K/F$, this definition agrees with the previous one, and moreover the formula

$${\mathrm{N}}_{F}^{K}(x):=\prod _{\sigma \in \mathrm{Gal}(K/F)}\sigma (x)$$ |

holds.

The norm of $x$ is always an element of $F$, since any element of $\mathrm{Gal}(K/F)$ permutes the orbit of $x$ and thus fixes ${\mathrm{N}}_{F}^{K}(x)$.

Title | norm |
---|---|

Canonical name | Norm |

Date of creation | 2013-03-22 12:18:02 |

Last modified on | 2013-03-22 12:18:02 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 5 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 12F05 |