Defining polynomial
|
\(x^{20} - 2 x + 7\)
|
Invariants
| Base field: | $\Q_{67}$ |
| Degree $d$: | $20$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $20$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{67}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 67 }) }$: | $20$ |
| This field is Galois and abelian over $\Q_{67}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{67}(\sqrt{2})$, 67.4.0.1, 67.5.0.1, 67.10.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 67.20.0.1 $\cong \Q_{67}(t)$ where $t$ is a root of
\( x^{20} - 2 x + 7 \)
|
| Relative Eisenstein polynomial: |
\( x - 67 \)
$\ \in\Q_{67}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
| Galois group: | $C_{20}$ (as 20T1) |
| Inertia group: | trivial |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $20$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: | Not computed |