Defining polynomial
| \( x^{15} + 3 x^{14} + 18 x^{12} + 9 x^{11} + 12 x^{10} + 10 x^{9} + 12 x^{8} + 6 x^{6} + 12 x^{5} + 12 x^{4} + 25 x^{3} + 21 x^{2} + 9 x + 1 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $15$ |
| Ramification exponent $e$ : | $3$ |
| Residue field degree $f$ : | $5$ |
| Discriminant exponent $c$ : | $15$ |
| 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
| 3.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.5.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{5} - x + 1 \) |
| Relative Eisenstein polynomial: | $ x^{3} + \left(3 t + 6\right) x^{2} + \left(6 t^{4} + 6 t^{2} + 3\right) x + 6 t^{4} + 3 t^{3} + 6 t^{2} + 6 t \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | 15T33 |
| Inertia group: | Intransitive group isomorphic to $C_3:(C_3^3:C_2)$ |
| Unramified degree: | $5$ |
| Tame degree: | $2$ |
| Wild slopes: | [3/2, 3/2, 3/2, 3/2] |
| Galois mean slope: | $241/162$ |
| Galois splitting model: | $x^{15} + 6 x^{13} - 12 x^{12} - 45 x^{11} - 51 x^{10} - 3 x^{9} + 225 x^{8} + 435 x^{7} - 4 x^{6} - 585 x^{5} - 216 x^{4} + 403 x^{3} + 135 x^{2} - 201 x + 43$ |