Defining polynomial
| \( x^{15} + 15 x^{14} + 24 x^{13} + 6 x^{12} + 21 x^{11} + 3 x^{10} + x^{9} + 3 x^{8} + 15 x^{7} + 9 x^{6} + 18 x^{5} + 25 x^{3} + 21 x^{2} + 9 x + 4 \) |
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: | $i$ |
| $|\Aut(K/\Q_{ 3 })|$: | $5$ |
| This field is not Galois over $\Q_{3}$. | |
Intermediate fields
| 3.3.3.2, 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(6 t^{4} + 3 t^{2} + 3 t\right) x^{2} + \left(3 t^{3} + 6 t^{2} + 6\right) x + 6 t^{4} + 3 t^{3} + 6 t^{2} + 6 t + 6 \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_5\times S_3$ (as 15T4) |
| Inertia group: | Intransitive group isomorphic to $S_3$ |
| Unramified degree: | $5$ |
| Tame degree: | $2$ |
| Wild slopes: | [3/2] |
| Galois mean slope: | $7/6$ |
| Galois splitting model: | $x^{15} - 3 x^{14} + 18 x^{13} - 13 x^{12} + 378 x^{11} - 678 x^{10} + 2363 x^{9} + 966 x^{8} + 9105 x^{7} - 9552 x^{6} + 5115 x^{5} + 291 x^{4} - 768 x^{3} + 63 x^{2} + 18 x - 1$ |