## Defining polynomial

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

## Invariants

Base field: | $\Q_{5}$ |

Degree $d$ : | $2$ |

Ramification exponent $e$ : | $1$ |

Residue field degree $f$ : | $2$ |

Discriminant exponent $c$ : | $0$ |

Discriminant root field: | $\Q_{5}(\sqrt{*})$ |

Root number: | $1$ |

$|\Gal(K/\Q_{ 5 })|$: | $2$ |

This field is Galois and abelian over $\Q_{5}$. |

## Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 5 }$. |

## Unramified/totally ramified tower

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

Relative Eisenstein polynomial: | $ x - 5 \in\Q_{5}(t)[x]$ |

## Invariants of the Galois closure

Galois group: | $C_2$ |

Inertia group: | Trivial |

Unramified degree: | $2$ |

Tame degree: | $1$ |

Wild slopes: | None |

Galois Mean Slope: | $0$ |

Global Splitting Model: | \( x^{2} - x + 2 \) |