Defining polynomial
| \( x^{9} + 6 x^{6} + 18 x^{5} + 36 x^{3} + 27 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $9$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 3 })|$: | $3$ |
| 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(18 t^{2} + 3 t + 6\right) x^{2} + 18 t^{2} x + 3 t^{2} + 18 t + 9 \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $He_3$ (as 9T7) |
| Inertia group: | Intransitive group isomorphic to $C_3^2$ |
| Unramified degree: | $3$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2] |
| Galois mean slope: | $16/9$ |
| Galois splitting model: | $x^{9} - 24 x^{7} - 23 x^{6} + 135 x^{5} + 159 x^{4} - 209 x^{3} - 180 x^{2} + 141 x + 11$ |