# Galois group

The *Galois group ^{}* $\mathrm{Gal}(K/F)$ of a field extension $K/F$ is the group of all field automorphisms $\sigma :K\to K$ of $K$ which fix $F$ (i.e., $\sigma (x)=x$ for all $x\in F$). The group operation

^{}is given by composition: for two automorphisms

^{}${\sigma}_{1},{\sigma}_{2}\in \mathrm{Gal}(K/F)$, given by ${\sigma}_{1}:K\to K$ and ${\sigma}_{2}:K\to K$, the product

^{}${\sigma}_{1}\cdot {\sigma}_{2}\in \mathrm{Gal}(K/F)$ is the composite of the two maps ${\sigma}_{1}\circ {\sigma}_{2}:K\to K$.

The *Galois group* of a polynomial^{} $f(x)\in F[x]$ is defined to be the Galois group of the splitting field^{} of $f(x)$ over $F$.

Title | Galois group |
---|---|

Canonical name | GaloisGroup |

Date of creation | 2013-03-22 12:08:19 |

Last modified on | 2013-03-22 12:08:19 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 8 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 12F10 |

Related topic | FundamentalTheoremOfGaloisTheory |

Related topic | InfiniteGaloisTheory |