## Defining polynomial

\(x^{17} + 2\) |

## Invariants

Base field: | $\Q_{2}$ |

Degree $d$: | $17$ |

Ramification exponent $e$: | $17$ |

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

Discriminant exponent $c$: | $16$ |

Discriminant root field: | $\Q_{2}$ |

Root number: | $1$ |

$\card{ \Aut(K/\Q_{ 2 }) }$: | $1$ |

This field is not Galois over $\Q_{2}.$ | |

Visible slopes: | None |

## Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 2 }$. |

## Unramified/totally ramified tower

Unramified subfield: | $\Q_{2}$ |

Relative Eisenstein polynomial: | \( x^{17} + 2 \) |

## Ramification polygon

Residual polynomials: | $z^{16} + z^{15} + 1$ |

Associated inertia: | $8$ |

Indices of inseparability: | $[0]$ |

## Invariants of the Galois closure

Galois group: | $C_{17}:C_8$ (as 17T4) |

Inertia group: | $C_{17}$ (as 17T1) |

Wild inertia group: | $C_1$ |

Unramified degree: | $8$ |

Tame degree: | $17$ |

Wild slopes: | None |

Galois mean slope: | $16/17$ |

Galois splitting model: | Not computed |