## Defining polynomial

\(x^{8} + 9\) |

## Invariants

Base field: | $\Q_{3}$ |

Degree $d$: | $8$ |

Ramification exponent $e$: | $4$ |

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

Discriminant exponent $c$: | $6$ |

Discriminant root field: | $\Q_{3}$ |

Root number: | $1$ |

$\card{ \Gal(K/\Q_{ 3 }) }$: | $8$ |

This field is Galois over $\Q_{3}.$ | |

Visible slopes: | None |

## Intermediate fields

$\Q_{3}(\sqrt{2})$, $\Q_{3}(\sqrt{3})$, $\Q_{3}(\sqrt{3\cdot 2})$, 3.4.2.1 |

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Unramified/totally ramified tower

Unramified subfield: | $\Q_{3}(\sqrt{2})$ $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{2} + 2 x + 2 \) |

Relative Eisenstein polynomial: | \( x^{4} + 3 t + 3 \) $\ \in\Q_{3}(t)[x]$ |

## Ramification polygon

Residual polynomials: | $z^{3} + z^{2} + 1$ |

Associated inertia: | $1$ |

Indices of inseparability: | $[0]$ |

## Invariants of the Galois closure

Galois group: | $Q_8$ (as 8T5) |

Inertia group: | Intransitive group isomorphic to $C_4$ |

Wild inertia group: | $C_1$ |

Unramified degree: | $2$ |

Tame degree: | $4$ |

Wild slopes: | None |

Galois mean slope: | $3/4$ |

Galois splitting model: | $x^{8} + 12 x^{6} + 36 x^{4} + 36 x^{2} + 9$ |

# Additional information

There is a tamely ramified extension of $\Q_p$ with Galois group $Q_8$, the quaternion group, if and only if $p\equiv 3 \pmod4$, and for each such $p$, there is a unique extension. This is the first such example, in that is involves the smallest such odd prime $p$.