Defining polynomial
|
\(x^{8} + 2 x^{5} + x^{4} + 2 x^{2} + 2 x + 2\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $8$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 3 }) }$: | $8$ |
| This field is Galois and abelian over $\Q_{3}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.4.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.8.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{8} + 2 x^{5} + x^{4} + 2 x^{2} + 2 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x - 3 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
| Galois group: | $C_8$ (as 8T1) |
| Inertia group: | trivial |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $8$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: | $x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{2} + 2$ |