Defining polynomial
| \( x^{12} + 3 x^{10} - 3 x^{9} - 3 x^{7} + 3 x^{6} + 3 x^{5} + 3 x^{4} + 3 x^{3} - 3 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $15$ |
| 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
| $\Q_{3}(\sqrt{3})$, 3.3.3.2, 3.4.3.2, 3.6.7.2 |
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: | \( x^{12} + 3 x^{10} - 3 x^{9} - 3 x^{7} + 3 x^{6} + 3 x^{5} + 3 x^{4} + 3 x^{3} - 3 \) |