Defining polynomial
| \( x^{8} + 2 x^{4} + 8 x + 12 \) |
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: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{*})$, 2.4.8.7 |
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} + 4 x + 6 t + 6 \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: | [8/3, 8/3] |
| Galois mean slope: | $13/6$ |
| Galois splitting model: | $x^{8} + 2 x^{6} - 8 x^{5} + 13 x^{4} - 8 x^{3} + 6 x^{2} - 4 x + 3$ |