## Defining polynomial

\(x^{2} + 4 x + 2\) |

## Invariants

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

Degree $d$: | $2$ |

Ramification exponent $e$: | $1$ |

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

Discriminant exponent $c$: | $0$ |

Discriminant root field: | $\Q_{5}(\sqrt{2})$ |

Root number: | $1$ |

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

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

Visible slopes: | None |

## Intermediate fields

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

## Unramified/totally ramified tower

Unramified subfield: | $\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{2} + 4 x + 2 \) |

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

## Ramification polygon

The ramification polygon is trivial for unramified extensions.

## Invariants of the Galois closure

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

Inertia group: | trivial |

Wild inertia group: | $C_1$ |

Unramified degree: | $2$ |

Tame degree: | $1$ |

Wild slopes: | None |

Galois mean slope: | $0$ |

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