# Properties

 Label 2.8.8.11 Base $$\Q_{2}$$ Degree $$8$$ e $$4$$ f $$2$$ c $$8$$ Galois group $S_4$ (as 8T14)

# Related objects

## Defining polynomial

 $$x^{8} + 20 x^{2} + 4$$

## Invariants

 Base field: $\Q_{2}$ Degree $d$ : $8$ Ramification exponent $e$ : $4$ Residue field degree $f$ : $2$ Discriminant exponent $c$ : $8$ Discriminant root field: $\Q_{2}$ Root number: $1$ $|\Aut(K/\Q_{ 2 })|$: $2$ This field is not 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: $x^{4} + \left(4 t + 6\right) x^{3} + 2 x^{2} + \left(4 t + 6\right) x + 2 \in\Q_{2}(t)[x]$

## Invariants of the Galois closure

 Galois group: $S_4$ (as 8T14) Inertia group: Intransitive group isomorphic to $A_4$ Unramified degree: $2$ Tame degree: $3$ Wild slopes: [4/3, 4/3] Galois mean slope: $7/6$ Galois splitting model: $x^{8} - 4 x^{7} + 4 x^{6} - 2 x^{5} + 5 x^{4} - 2 x^{3} + 4 x^{2} - 4 x + 1$