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