Defining polynomial
| \( x^{8} + 8 x^{4} + 496 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $8$ |
| Ramification exponent $e$ : | $8$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $28$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $-i$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{2})$, 2.4.10.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: | \( x^{8} - 8 x^{7} + 28 x^{6} - 56 x^{5} + 520 x^{4} - 1856 x^{3} + 2728 x^{2} - 1808 x + 1569826 \) |
Invariants of the Galois closure
| Galois group: | $SD_{16}$ (as 8T8) |
| Inertia group: | $QD_{16}$ |
| Unramified degree: | $1$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 3, 7/2, 9/2] |
| Galois mean slope: | $29/8$ |
| Galois splitting model: | $x^{8} - 2 x^{4} - 1$ |