Defining polynomial
| \( x^{8} - 7 x^{4} + 147 \) |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 7 })|$: | $4$ |
| This field is not Galois over $\Q_{7}.$ | |
Intermediate fields
| $\Q_{7}(\sqrt{3})$, 7.4.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} - x + 3 \) |
| Relative Eisenstein polynomial: | $ x^{4} - 7 t \in\Q_{7}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $OD_{16}$ (as 8T7) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Unramified degree: | $4$ |
| Tame degree: | $4$ |
| Wild slopes: | None |
| Galois mean slope: | $3/4$ |
| Galois splitting model: | $x^{8} - x^{7} - 13 x^{6} - 13 x^{5} + 25 x^{4} + 38 x^{3} - 33 x^{2} - 34 x + 11$ |