Defining polynomial
\(x^{12} - 17\)  |
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $11$ |
| Discriminant root field: | $\Q_{17}(\sqrt{17})$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 17 })|$: | $4$ |
| This field is not Galois over $\Q_{17}.$ | |
Intermediate fields
| $\Q_{17}(\sqrt{17})$, 17.3.2.1, 17.4.3.1, 17.6.5.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{17}$ |
| Relative Eisenstein polynomial: | \( x^{12} - 17 \) |
Invariants of the Galois closure
| Galois group: | $C_4\times S_3$ (as 12T11) |
| Inertia group: | $C_{12}$ |
| Unramified degree: | $2$ |
| Tame degree: | $12$ |
| Wild slopes: | None |
| Galois mean slope: | $11/12$ |
| Galois splitting model: | Not computed |