## Defining polynomial

\( x^{3} - x + 1 \) |

## Invariants

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

Degree $d$ : | $3$ |

Ramification exponent $e$ : | $1$ |

Residue field degree $f$ : | $3$ |

Discriminant exponent $c$ : | $0$ |

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

Root number: | $1$ |

$|\Gal(K/\Q_{ 2 })|$: | $3$ |

This field is Galois and abelian over $\Q_{2}$. |

## Intermediate fields

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

## Unramified/totally ramified tower

Unramified subfield: | 2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{3} - x + 1 \) |

Relative Eisenstein polynomial: | $ x - 2 \in\Q_{2}(t)[x]$ |

## Invariants of the Galois closure

Galois group: | $C_3$ (as 3T1) |

Inertia group: | Trivial |

Unramified degree: | $3$ |

Tame degree: | $1$ |

Wild slopes: | None |

Galois mean slope: | $0$ |

Galois splitting model: | \( x^{3} - x^{2} - 2 x + 1 \) |