Defining polynomial
| \( x^{8} - 12167 x^{2} + 3078251 \) |
Invariants
| Base field: | $\Q_{23}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{23}(\sqrt{5})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 23 })|$: | $8$ |
| This field is Galois and abelian over $\Q_{23}.$ | |
Intermediate fields
| $\Q_{23}(\sqrt{5})$, 23.4.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 23.4.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of \( x^{4} - x + 11 \) |
| Relative Eisenstein polynomial: | $ x^{2} - 23 t \in\Q_{23}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_8$ (as 8T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Unramified degree: | $4$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | Not computed |