## Defining polynomial

\( x^{5} + 20 x + 5 \) |

## Invariants

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

Degree $d$: | $5$ |

Ramification exponent $e$: | $5$ |

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

Discriminant exponent $c$: | $5$ |

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

Root number: | $-1$ |

$|\Aut(K/\Q_{ 5 })|$: | $1$ |

This field is not Galois over $\Q_{5}.$ |

## Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 5 }$. |

## Unramified/totally ramified tower

Unramified subfield: | $\Q_{5}$ |

Relative Eisenstein polynomial: | \( x^{5} + 20 x + 5 \) |

## Invariants of the Galois closure

Galois group: | $F_5$ (as 5T3) |

Inertia group: | $F_5$ |

Unramified degree: | $1$ |

Tame degree: | $4$ |

Wild slopes: | [5/4] |

Galois mean slope: | $23/20$ |

Galois splitting model: | $x^{5} - 5 x^{3} + 5 x - 4$ |