## Defining polynomial

\( x^{4} - 5 x^{2} + 50 \) |

## Invariants

Base field: | $\Q_{5}$ |

Degree $d$ : | $4$ |

Ramification exponent $e$ : | $2$ |

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

Discriminant exponent $c$ : | $2$ |

Discriminant root field: | $\Q_{5}(\sqrt{*})$ |

Root number: | $1$ |

$|\Gal(K/\Q_{ 5 })|$: | $4$ |

This field is Galois and abelian over $\Q_{5}$. |

## Intermediate fields

$\Q_{5}(\sqrt{*})$ |

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

## Unramified/totally ramified tower

Unramified subfield: | $\Q_{5}(\sqrt{*})$ $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{2} - x + 2 \) |

Relative Eisenstein polynomial: | $ x^{2} - 5 t \in\Q_{5}(t)[x]$ |

## Invariants of the Galois closure

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

Inertia group: | Intransitive group isomorphic to $C_2$ |

Unramified degree: | $2$ |

Tame degree: | $2$ |

Wild slopes: | None |

Galois mean slope: | $1/2$ |

Galois splitting model: | $x^{4} + 20 x^{2} + 50$ |