Defining polynomial
|
\(x^{8} + 65 x^{3} + x^{2} + 32 x + 5\)
|
Invariants
| Base field: | $\Q_{97}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $8$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{97}(\sqrt{5})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 97 }) }$: | $8$ |
| This field is Galois and abelian over $\Q_{97}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{97}(\sqrt{5})$, 97.4.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 97.8.0.1 $\cong \Q_{97}(t)$ where $t$ is a root of
\( x^{8} + 65 x^{3} + x^{2} + 32 x + 5 \)
|
| Relative Eisenstein polynomial: |
\( x - 97 \)
$\ \in\Q_{97}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.