## Defining polynomial

\( x^{8} + x^{2} - 2 x + 6 \) |

## Invariants

Base field: | $\Q_{11}$ |

Degree $d$ : | $8$ |

Ramification exponent $e$ : | $1$ |

Residue field degree $f$ : | $8$ |

Discriminant exponent $c$ : | $0$ |

Discriminant root field: | $\Q_{11}(\sqrt{*})$ |

Root number: | $1$ |

$|\Gal(K/\Q_{ 11 })|$: | $8$ |

This field is Galois and abelian over $\Q_{11}$. |

## Intermediate fields

$\Q_{11}(\sqrt{*})$, 11.4.0.1 |

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

## Unramified/totally ramified tower

Unramified subfield: | 11.8.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{8} + x^{2} - 2 x + 6 \) |

Relative Eisenstein polynomial: | $ x - 11 \in\Q_{11}(t)[x]$ |