This field has $7$ quadratic subextensions (the most among all $p$-adic fields), which is used in the hardest case of the Kronecker-Weber theorem ($p=2$). It is also the splitting field over $\mathbb{Q}_2$ of $x^8-16$, the counterexample that corrected the statement of the Grunwald-Wang theorem.

## Defining polynomial

\(x^{8} + 4 x^{6} + 8 x^{2} + 4\) |

## Invariants

Base field: | $\Q_{2}$ |

Degree $d$: | $8$ |

Ramification exponent $e$: | $4$ |

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

Discriminant exponent $c$: | $16$ |

Discriminant root field: | $\Q_{2}$ |

Root number: | $1$ |

$|\Gal(K/\Q_{ 2 })|$: | $8$ |

This field is Galois and abelian over $\Q_{2}.$ |

## Intermediate fields

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

## Unramified/totally ramified tower

Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} - x + 1 \) |

Relative Eisenstein polynomial: | \( x^{4} + \left(8 t + 4\right) x^{3} + 10 x^{2} + \left(8 t + 12\right) x + 14 \)$\ \in\Q_{2}(t)[x]$ |

## Invariants of the Galois closure

Galois group: | $C_2^3$ (as 8T3) |

Inertia group: | Intransitive group isomorphic to $C_2^2$ |

Unramified degree: | $2$ |

Tame degree: | $1$ |

Wild slopes: | [2, 3] |

Galois mean slope: | $2$ |

Galois splitting model: | $x^{8} - x^{4} + 1$ |