## Defining polynomial

\( x^{11} + 736 \) |

## Invariants

Base field: | $\Q_{23}$ |

Degree $d$ : | $11$ |

Ramification exponent $e$ : | $11$ |

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

Discriminant exponent $c$ : | $10$ |

Discriminant root field: | $\Q_{23}$ |

Root number: | $1$ |

$|\Gal(K/\Q_{ 23 })|$: | $11$ |

This field is Galois and abelian over $\Q_{23}$. |

## Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 23 }$. |

## Unramified/totally ramified tower

Unramified subfield: | $\Q_{23}$ |

Relative Eisenstein polynomial: | \( x^{11} + 736 \) |

## Invariants of the Galois closure

Galois group: | $C_{11}$ (as 11T1) |

Inertia group: | $C_{11}$ |

Unramified degree: | $1$ |

Tame degree: | $11$ |

Wild slopes: | None |

Galois mean slope: | $10/11$ |

Galois splitting model: | Not computed |