## Defining polynomial

\(x^{5} + 2 x + 65\) |

## Invariants

Base field: | $\Q_{67}$ |

Degree $d$: | $5$ |

Ramification exponent $e$: | $1$ |

Residue field degree $f$: | $5$ |

Discriminant exponent $c$: | $0$ |

Discriminant root field: | $\Q_{67}$ |

Root number: | $1$ |

$\card{ \Gal(K/\Q_{ 67 }) }$: | $5$ |

This field is Galois and abelian over $\Q_{67}.$ | |

Visible slopes: | None |

## Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 67 }$. |

## Unramified/totally ramified tower

Unramified subfield: | 67.5.0.1 $\cong \Q_{67}(t)$ where $t$ is a root of \( x^{5} + 2 x + 65 \) |

Relative Eisenstein polynomial: | \( x - 67 \) $\ \in\Q_{67}(t)[x]$ |

## Ramification polygon

The ramification polygon is trivial for unramified extensions.