## Defining polynomial

\(x^{4} - x + 30\) |

## Invariants

Base field: | $\Q_{109}$ |

Degree $d$: | $4$ |

Ramification exponent $e$: | $1$ |

Residue field degree $f$: | $4$ |

Discriminant exponent $c$: | $0$ |

Discriminant root field: | $\Q_{109}(\sqrt{2})$ |

Root number: | $1$ |

$|\Gal(K/\Q_{ 109 })|$: | $4$ |

This field is Galois and abelian over $\Q_{109}.$ |

## Intermediate fields

$\Q_{109}(\sqrt{2})$ |

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Unramified/totally ramified tower

Unramified subfield: | 109.4.0.1 $\cong \Q_{109}(t)$ where $t$ is a root of \( x^{4} - x + 30 \) |

Relative Eisenstein polynomial: | \( x - 109 \)$\ \in\Q_{109}(t)[x]$ |

## Invariants of the Galois closure

Galois group: | $C_4$ (as 4T1) |

Inertia group: | trivial |

Unramified degree: | $4$ |

Tame degree: | $1$ |

Wild slopes: | None |

Galois mean slope: | $0$ |

Galois splitting model: | $x^{4} - x^{3} + 2 x^{2} + 4 x + 3$ |