Defining polynomial
| \( x^{8} + 8 x^{3} + 14 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $8$ |
| Ramification exponent $e$ : | $8$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $26$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $1$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{2})$ |
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: | \( x^{8} + 8 x^{3} + 14 \) |
Invariants of the Galois closure
| Galois group: | $C_2^2:S_4:C_2$ (as 8T41) |
| Inertia group: | $C_2^4:C_6$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [8/3, 8/3, 3, 23/6, 23/6] |
| Galois mean slope: | $169/48$ |
| Galois splitting model: | $x^{8} + 4 x^{6} - 16 x^{5} - 114 x^{4} + 8 x^{3} + 668 x^{2} - 696 x + 191$ |