Defining polynomial
| \( x^{8} + x^{2} - 2 x + 5 \) |
Invariants
| Base field: | $\Q_{23}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $8$ |
| Discriminant exponent $c$: | $0$ |
| 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: | 23.8.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of \( x^{8} + x^{2} - 2 x + 5 \) |
| Relative Eisenstein polynomial: | $ x - 23 \in\Q_{23}(t)[x]$ |