Defining polynomial
|
\(x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 3 }) }$: | $8$ |
| This field is Galois over $\Q_{3}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, $\Q_{3}(\sqrt{3})$, $\Q_{3}(\sqrt{3\cdot 2})$, 3.4.2.1, 3.4.3.1 x2, 3.4.3.2 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}(\sqrt{2})$ $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{2} + 2 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + \left(3 t + 6\right) x + 3 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{3} + z^{2} + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $D_4$ (as 8T4) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $4$ |
| Wild slopes: | None |
| Galois mean slope: | $3/4$ |
| Galois splitting model: | $x^{8} - 3 x^{7} + 4 x^{6} - 3 x^{5} + 3 x^{4} - 3 x^{3} + 4 x^{2} - 3 x + 1$ |