Defining polynomial
| \( x^{6} + 3 x^{4} + 6 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $6$ |
| Ramification exponent $e$ : | $6$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $9$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| Root number: | $i$ |
| $|\Gal(K/\Q_{ 3 })|$: | $6$ |
| This field is Galois and abelian over $\Q_{3}$. | |
Intermediate fields
| $\Q_{3}(\sqrt{3})$, 3.3.4.1 |
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: | \( x^{6} + 3 x^{4} + 6 \) |
Invariants of the Galois closure
| Galois group: | $C_6$ (as 6T1) |
| Inertia group: | $C_6$ |
| Unramified degree: | $1$ |
| Tame degree: | $2$ |
| Wild slopes: | [2] |
| Galois mean slope: | $3/2$ |
| Galois splitting model: | $x^{6} - 42 x^{4} + 441 x^{2} - 588$ |