Defining polynomial
| \( x^{8} + 8 x^{6} + 8 x^{2} + 20 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $8$ |
| Ramification exponent $e$ : | $8$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $16$ |
| Discriminant root field: | $\Q_{2}(\sqrt{*})$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.4.6.7 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: | \( x^{8} + 24 x^{7} + 31754 x^{6} - 7980 x^{5} + 93242 x^{4} + 131040 x^{3} + 1983672 x^{2} + 1922940 x + 5996430 \) |
Invariants of the Galois closure
| Galois group: | $C_2\wr A_4$ (as 8T38) |
| Inertia group: | $C_2^3 : D_4 $ |
| Unramified degree: | $6$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2, 2, 2, 5/2] |
| Galois mean slope: | $35/16$ |
| Galois splitting model: | $x^{8} - 4 x^{7} + 4 x^{5} + 22 x^{4} - 4 x^{3} + 4 x + 1$ |