Defining polynomial
| \( x^{14} - 3 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $13$ |
| 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
| $\Q_{3}(\sqrt{3})$, 3.7.6.1 |
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: | \( x^{14} - 3 \) |
Invariants of the Galois closure
| Galois group: | $C_2\times F_7$ (as 14T7) |
| Inertia group: | $C_{14}$ |
| Unramified degree: | $6$ |
| Tame degree: | $14$ |
| Wild slopes: | None |
| Galois mean slope: | $13/14$ |
| Galois splitting model: | Not computed |