## Defining polynomial

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

## Invariants

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

Degree $d$: | $5$ |

Ramification exponent $e$: | $1$ |

Residue field degree $f$: | $5$ |

Discriminant exponent $c$: | $0$ |

Discriminant root field: | $\Q_{11}$ |

Root number: | $1$ |

$|\Gal(K/\Q_{ 11 })|$: | $5$ |

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

## Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 11 }$. |

## Unramified/totally ramified tower

Unramified subfield: | 11.5.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{5} + x^{2} - x + 5 \) |

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

## Invariants of the Galois closure

Galois group: | $C_5$ (as 5T1) |

Inertia group: | trivial |

Unramified degree: | $5$ |

Tame degree: | $1$ |

Wild slopes: | None |

Galois mean slope: | $0$ |

Galois splitting model: | $x^{5} - 10 x^{3} - 5 x^{2} + 10 x - 1$ |