Defining polynomial
| \( x^{8} + 8 x^{5} + 4 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $8$ |
| Ramification exponent $e$ : | $8$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $16$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $1$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$ |
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: | \( x^{8} - 4 x^{7} - 16 x^{6} - 8 x^{5} + 98 x^{4} - 128 x^{3} + 76 x^{2} - 20 x + 2 \) |
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: | [4/3, 4/3, 2, 7/3, 7/3] |
| Galois mean slope: | $103/48$ |
| Galois splitting model: | $x^{8} - 8 x^{6} - 12 x^{5} + 24 x^{4} + 60 x^{3} + 64 x^{2} + 36 x + 13$ |