Defining polynomial
| \( x^{12} + 93 x^{11} + 351 x^{10} + 3 x^{9} + 126 x^{8} - 297 x^{7} + 171 x^{6} + 243 x^{5} - 324 x^{4} - 54 x^{3} + 162 x^{2} - 243 x + 324 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $3$ |
| Residue field degree $f$ : | $4$ |
| Discriminant exponent $c$ : | $16$ |
| 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, 3.4.0.1, 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: | 3.4.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{4} - x + 2 \) |
| Relative Eisenstein polynomial: | $ x^{3} + \left(6 t^{3} + 6 t^{2} - 12\right) x^{2} + \left(9 t^{3} - 9 t^{2}\right) x + 12 t^{3} - 3 t + 6 \in\Q_{3}(t)[x]$ |