Defining polynomial
| \( x^{12} + 21 x^{11} - 21 x^{10} + 21 x^{9} - 27 x^{7} + 15 x^{6} + 18 x^{5} - 27 x^{4} + 27 x^{3} + 27 x^{2} + 27 x - 36 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $6$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $18$ |
| Discriminant root field: | $\Q_{3}(\sqrt{*})$ |
| Root number: | $-1$ |
| $|\Gal(K/\Q_{ 3 })|$: | $12$ |
| This field is Galois over $\Q_{3}$. | |
Intermediate fields
| $\Q_{3}(\sqrt{*})$, 3.3.4.4 x3, 3.4.2.2, 3.6.8.5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}(\sqrt{*})$ $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{2} - x + 2 \) |
| Relative Eisenstein polynomial: | $ x^{6} + \left(-12 t + 12\right) x^{5} + \left(-12 t - 9\right) x^{4} + \left(12 t + 9\right) x^{3} - 9 x^{2} - 9 x - 6 t - 12 \in\Q_{3}(t)[x]$ |