Defining polynomial
\(x^{8} + 28 x^{4} + 144\)  |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 2 })|$: | $8$ |
| This field is Galois and abelian over $\Q_{2}.$ | |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{-5})$, 2.4.0.1, 2.4.4.1, 2.4.4.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.4.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{4} - x + 1 \) |
| Relative Eisenstein polynomial: | \( x^{2} + \left(2 t^{3} + 2 t^{2} + 2 t + 2\right) x + 2 t^{3} + 2 t \)$\ \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times C_4$ (as 8T2) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Unramified degree: | $4$ |
| Tame degree: | $1$ |
| Wild slopes: | [2] |
| Galois mean slope: | $1$ |
| Galois splitting model: | $x^{8} - x^{6} + x^{4} - x^{2} + 1$ |