## Defining polynomial

\( x^{11} - x + 16 \) |

## Invariants

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

Degree $d$ : | $11$ |

Ramification exponent $e$ : | $1$ |

Residue field degree $f$ : | $11$ |

Discriminant exponent $c$ : | $0$ |

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

Root number: | $1$ |

$|\Gal(K/\Q_{ 19 })|$: | $11$ |

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

## Intermediate fields

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

## Unramified/totally ramified tower

Unramified subfield: | 19.11.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{11} - x + 16 \) |

Relative Eisenstein polynomial: | $ x - 19 \in\Q_{19}(t)[x]$ |