# trace

Let $K/F$ be a Galois extension^{}, and let $x\in K$. The trace ${\mathrm{Tr}}_{F}^{K}(x)$ of $x$ is defined to be the sum 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,

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

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

The name “trace” derives from the fact that, when $K/F$ is finite, the trace of $x$ is simply the trace of the linear transformation $T:K\u27f6K$ of vector spaces^{} over $F$ defined by $T(v):=xv$.

Title | trace |
---|---|

Canonical name | Trace1 |

Date of creation | 2013-03-22 12:17:59 |

Last modified on | 2013-03-22 12:17:59 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 7 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 12F05 |