## Defining polynomial

\(x^{3} + 19\) |

## Invariants

Base field: | $\Q_{19}$ |

Degree $d$: | $3$ |

Ramification exponent $e$: | $3$ |

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

Discriminant exponent $c$: | $2$ |

Discriminant root field: | $\Q_{19}$ |

Root number: | $1$ |

$\card{ \Gal(K/\Q_{ 19 }) }$: | $3$ |

This field is Galois and abelian over $\Q_{19}.$ | |

Visible slopes: | None |

## Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 19 }$. |

## Unramified/totally ramified tower

Unramified subfield: | $\Q_{19}$ |

Relative Eisenstein polynomial: | \( x^{3} + 19 \) |

## Ramification polygon

Residual polynomials: | $z^{2} + 3z + 3$ |

Associated inertia: | $1$ |

Indices of inseparability: | $[0]$ |

## Invariants of the Galois closure

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

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

Wild inertia group: | $C_1$ |

Unramified degree: | $1$ |

Tame degree: | $3$ |

Wild slopes: | None |

Galois mean slope: | $2/3$ |

Galois splitting model: | $x^{3} - x^{2} - 6 x + 7$ |