## Defining polynomial

\( x^{2} - x + 3 \) |

## Invariants

Base field: | $\Q_{7}$ |

Degree $d$ : | $2$ |

Ramification exponent $e$ : | $1$ |

Residue field degree $f$ : | $2$ |

Discriminant exponent $c$ : | $0$ |

Discriminant root field: | $\Q_{7}(\sqrt{*})$ |

Root number: | $1$ |

$|\Gal(K/\Q_{ 7 })|$: | $2$ |

This field is Galois and abelian over $\Q_{7}$. |

## Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$. |

## Unramified/totally ramified tower

Unramified subfield: | $\Q_{7}(\sqrt{*})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} - x + 3 \) |

Relative Eisenstein polynomial: | $ x - 7 \in\Q_{7}(t)[x]$ |

## Invariants of the Galois closure

Galois group: | $C_2$ (as 2T1) |

Inertia group: | Trivial |

Unramified degree: | $2$ |

Tame degree: | $1$ |

Wild slopes: | None |

Galois mean slope: | $0$ |

Galois splitting model: | \( x^{2} - x + 3 \) |