## Defining polynomial

\(x^{2} + 41\) |

## Invariants

Base field: | $\Q_{41}$ |

Degree $d$: | $2$ |

Ramification exponent $e$: | $2$ |

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

Discriminant exponent $c$: | $1$ |

Discriminant root field: | $\Q_{41}(\sqrt{41})$ |

Root number: | $1$ |

$\card{ \Gal(K/\Q_{ 41 }) }$: | $2$ |

This field is Galois and abelian over $\Q_{41}.$ | |

Visible slopes: | None |

## Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 41 }$. |

## Unramified/totally ramified tower

Unramified subfield: | $\Q_{41}$ |

Relative Eisenstein polynomial: | \( x^{2} + 41 \) |

## Ramification polygon

Residual polynomials: | $z + 2$ |

Associated inertia: | $1$ |

Indices of inseparability: | $[0]$ |

## Invariants of the Galois closure

Galois group: | $C_2$ (as 2T1) |

Inertia group: | $C_2$ (as 2T1) |

Wild inertia group: | $C_1$ |

Unramified degree: | $1$ |

Tame degree: | $2$ |

Wild slopes: | None |

Galois mean slope: | $1/2$ |

Galois splitting model: | $x^{2} - 12 x - 5$ |