## Defining polynomial

\(x + 2\) |

## Invariants

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

Degree $d$: | $1$ |

Ramification exponent $e$: | $1$ |

Residue field degree $f$: | $1$ |

Discriminant exponent $c$: | $0$ |

Discriminant root field: | $\Q_{7}$ |

Root number: | $1$ |

$\card{ \Gal(K/\Q_{ 7 }) }$: | $1$ |

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

Visible slopes: | None |

## Intermediate fields

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

## Unramified/totally ramified tower

Unramified subfield: | $\Q_{7}$ |

Relative Eisenstein polynomial: | \( x - 7 \) |

Indices of inseparability: | $[0]$ |

## Invariants of the Galois closure

Galois group: | $C_1$ (as 1T1) |

Inertia group: | $C_1$ (as 1T1) |

Wild inertia group: | $C_1$ |

Unramified degree: | $1$ |

Tame degree: | $1$ |

Wild slopes: | None |

Galois mean slope: | $0$ |

Galois splitting model: | $x + 2$ |