# Properties

 Label 2.8.22.1 Base $$\Q_{2}$$ Degree $$8$$ e $$4$$ f $$2$$ c $$22$$ Galois group $D_4$ (as 8T4)

# Related objects

## Defining polynomial

 $$x^{8} + 8 x^{5} + 6 x^{4} + 16 x^{3} + 8 x^{2} + 12$$

## Invariants

 Base field: $\Q_{2}$ Degree $d$ : $8$ Ramification exponent $e$ : $4$ Residue field degree $f$ : $2$ Discriminant exponent $c$ : $22$ Discriminant root field: $\Q_{2}$ Root number: $1$ $|\Gal(K/\Q_{ 2 })|$: $8$ This field is Galois 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: $\Q_{2}(\sqrt{*})$ $\cong \Q_{2}(t)$ where $t$ is a root of $$x^{2} - x + 1$$ Relative Eisenstein polynomial: $x^{4} + 8 x^{3} + \left(8 t + 4\right) x^{2} + \left(8 t + 8\right) x + 2 t + 2 \in\Q_{2}(t)[x]$

## Invariants of the Galois closure

 Galois group: $D_4$ (as 8T4) Inertia group: Intransitive group isomorphic to $C_4$ Unramified degree: $2$ Tame degree: $1$ Wild slopes: [3, 4] Galois mean slope: $11/4$ Galois splitting model: $$x^{8} + 4 x^{6} + 10 x^{4} + 24 x^{2} + 36$$