Defining polynomial
| \( x^{8} + 8 x^{7} + 36 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $8$ |
| Ramification exponent $e$ : | $8$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $18$ |
| Discriminant root field: | $\Q_{2}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $4$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, 2.4.6.8 |
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: | \( x^{8} + 86004 x^{7} - 1092250 x^{6} + 5982716 x^{5} - 14529012 x^{4} + 17582140 x^{3} - 42687824 x^{2} - 9996120 x - 72328822 \) |
Invariants of the Galois closure
| Galois group: | $C_4\wr C_2$ (as 8T17) |
| Inertia group: | $Q_8$ |
| Unramified degree: | $4$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2, 3] |
| Galois mean slope: | $9/4$ |
| Galois splitting model: | $x^{8} + 2 x^{4} + 5$ |