Defining polynomial
| \( x^{15} - 9 x^{5} + 27 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $15$ |
| Ramification exponent $e$ : | $5$ |
| Residue field degree $f$ : | $3$ |
| Discriminant exponent $c$ : | $12$ |
| Discriminant root field: | $\Q_{3}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 3 })|$: | $3$ |
| This field is not Galois over $\Q_{3}$. | |
Intermediate fields
| 3.3.0.1, 3.5.4.1 |
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^{5} + 6 t + 3 \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times F_5$ (as 15T8) |
| Inertia group: | Intransitive group isomorphic to $C_5$ |
| Unramified degree: | $12$ |
| Tame degree: | $5$ |
| Wild slopes: | None |
| Galois mean slope: | $4/5$ |
| Galois splitting model: | $x^{15} - 48 x^{10} - 513 x^{5} + 27$ |