Defining polynomial
| \( x^{15} + 42 x^{14} + 27 x^{13} + 31 x^{12} + 45 x^{11} + 45 x^{10} + 41 x^{9} + 30 x^{8} + 42 x^{7} + 8 x^{6} + 75 x^{5} + 18 x^{4} + 42 x^{3} + 24 x^{2} + 51 x + 19 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $15$ |
| Ramification exponent $e$ : | $3$ |
| Residue field degree $f$ : | $5$ |
| Discriminant exponent $c$ : | $20$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 3 })|$: | $15$ |
| This field is Galois and abelian over $\Q_{3}$. | |
Intermediate fields
| 3.3.4.3, 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} + 24 t^{3} + 15 t^{2} + 6 t + 12\right) x^{2} + \left(9 t^{4} + 18 t^{3} + 18 t^{2} + 9 t + 18\right) x + 9 t^{4} + 15 t^{3} + 15 t^{2} + 3 \in\Q_{3}(t)[x]$ |