Defining polynomial
| \( x^{9} + 9 x^{6} + 72 x^{3} + 27 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $9$ |
| Ramification exponent $e$ : | $3$ |
| Residue field degree $f$ : | $3$ |
| Discriminant exponent $c$ : | $15$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3*})$ |
| Root number: | $-i$ |
| $|\Aut(K/\Q_{ 3 })|$: | $3$ |
| This field is not Galois over $\Q_{3}$. | |
Intermediate fields
| 3.3.0.1, 3.3.5.2 |
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} + 18 t^{2} + 24 t \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times S_3$ (as 9T4) |
| Inertia group: | Intransitive group isomorphic to $S_3$ |
| Unramified degree: | $3$ |
| Tame degree: | $2$ |
| Wild slopes: | [5/2] |
| Galois mean slope: | $11/6$ |
| Galois splitting model: | $x^{9} - 12 x^{6} + 27 x^{3} + 27$ |