Defining polynomial
| \( x^{8} + 152 x^{4} + 16 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $8$ |
| Ramification exponent $e$ : | $8$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $22$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 2 })|$: | $8$ |
| This field is Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{-2*})$, $\Q_{2}(\sqrt{2*})$, 2.4.8.1, 2.4.9.4 x2, 2.4.10.4 x2 |
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: | \( x^{8} + 4 x^{7} + 756 x^{6} + 76432 x^{5} + 2054178 x^{4} - 9793880 x^{3} + 16455612 x^{2} - 11794856 x + 3080954 \) |
Invariants of the Galois closure
| Galois group: | $D_4$ (as 8T4) |
| Inertia group: | $D_4$ |
| Unramified degree: | $1$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 3, 7/2] |
| Galois mean slope: | $11/4$ |
| Galois splitting model: | $x^{8} + 152 x^{4} + 16$ |