# Properties

 Label 2.8.16.6 Base $$\Q_{2}$$ Degree $$8$$ e $$4$$ f $$2$$ c $$16$$ Galois group $C_2^3$ (as 8T3)

# Learn more about

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: $x^{8} - x^{4} + 1$