Defining polynomial
|
\(x^{9} + 9 x^{8} + 27 x^{7} + 36 x^{6} + 54 x^{5} + 81 x^{4} + 675 x^{3} + 2025 x^{2} - 3861\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $9$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $3$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| 3.3.0.1, 3.3.4.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.3.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{3} + 2 x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + 3 x^{2} + 18 t + 3 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{2} + 1$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times S_3$ (as 9T4) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_3$ |
| Unramified degree: | $6$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2]$ |
| Galois mean slope: | $4/3$ |
| Galois splitting model: | $x^{9} + 6 x^{7} - 8 x^{6} - 9 x^{5} - 60 x^{4} - 15 x^{3} - 18 x^{2} + 108 x + 8$ |