Defining polynomial
\(x^{12} - x^{4} - x^{3} - x^{2} + x - 1\)  |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $12$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 3 })|$: | $12$ |
| This field is Galois and abelian over $\Q_{3}.$ | |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.3.0.1, 3.4.0.1, 3.6.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.12.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{12} + 2 x^{4} - x^{3} + 2 x^{2} - 2 x + 2 \) |
| Relative Eisenstein polynomial: | \( x - 3 \)$\ \in\Q_{3}(t)[x]$ |