Defining polynomial
\(x^{7} - x + 8\)  |
Invariants
| Base field: | $\Q_{23}$ |
| Degree $d$: | $7$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $7$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{23}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 23 })|$: | $7$ |
| This field is Galois and abelian over $\Q_{23}.$ | |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 23 }$. |
Unramified/totally ramified tower
| Unramified subfield: | 23.7.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of \( x^{7} - x + 8 \) |
| Relative Eisenstein polynomial: | \( x - 23 \)$\ \in\Q_{23}(t)[x]$ |