Defining polynomial
| \( x^{12} - 12 x^{11} - 3 x^{10} + 6 x^{9} + 9 x^{7} + 6 x^{6} - 9 x^{5} - 9 x^{4} + 6 x^{3} + 9 x^{2} + 9 x - 6 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $21$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| Root number: | $-i$ |
| $|\Aut(K/\Q_{ 3 })|$: | $2$ |
| This field is not Galois over $\Q_{3}$. | |
Intermediate fields
| $\Q_{3}(\sqrt{3*})$, 3.4.3.1, 3.6.10.8 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}$ |
| Relative Eisenstein polynomial: | \( x^{12} - 12 x^{11} - 3 x^{10} + 6 x^{9} + 9 x^{7} + 6 x^{6} - 9 x^{5} - 9 x^{4} + 6 x^{3} + 9 x^{2} + 9 x - 6 \) |