Defining polynomial
| \( x^{9} + 3 x^{7} + 3 x^{6} + 54 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $9$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| Root number: | $i$ |
| $|\Aut(K/\Q_{ 3 })|$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
Intermediate fields
| 3.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.3.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{3} - x + 1 \) |
| Relative Eisenstein polynomial: | $ x^{3} + \left(6 t^{2} + 3\right) x + 6 t^{2} + 3 \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_3^3:C_6$ (as 9T22) |
| Inertia group: | Intransitive group isomorphic to $C_3^3:C_2$ |
| Unramified degree: | $3$ |
| Tame degree: | $2$ |
| Wild slopes: | [3/2, 3/2, 3/2] |
| Galois mean slope: | $79/54$ |
| Galois splitting model: | $x^{9} + 6 x^{7} - 9 x^{6} - 9 x^{5} - 15 x^{4} - 7 x^{3} + 27 x^{2} + 12 x - 13$ |