Defining polynomial
| \( x^{8} + 44 x^{4} + 100 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $8$ |
| Ramification exponent $e$ : | $8$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $24$ |
| 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, 2.4.11.20 x2, 2.4.11.11 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: | \( x^{8} - 8 x^{7} + 24 x^{6} - 32 x^{5} + 10 x^{4} + 24 x^{3} - 132 x^{2} + 216 x + 522 \) |
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, 4] |
| Galois mean slope: | $3$ |
| Galois splitting model: | $x^{8} + 4 x^{6} + 18 x^{4} - 68 x^{2} + 49$ |