## Defining polynomial

\(x^{6} - x + 2\) |

## Invariants

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

Degree $d$: | $6$ |

Ramification exponent $e$: | $1$ |

Residue field degree $f$: | $6$ |

Discriminant exponent $c$: | $0$ |

Discriminant root field: | $\Q_{5}(\sqrt{2})$ |

Root number: | $1$ |

$|\Gal(K/\Q_{ 5 })|$: | $6$ |

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

## Intermediate fields

$\Q_{5}(\sqrt{2})$, 5.3.0.1 |

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

## Unramified/totally ramified tower

Unramified subfield: | 5.6.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{6} - x + 2 \) |

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