Defining polynomial
| \( x^{14} - 11 \) |
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $13$ |
| Discriminant root field: | $\Q_{11}(\sqrt{11})$ |
| Root number: | $i$ |
| $|\Aut(K/\Q_{ 11 })|$: | $2$ |
| This field is not Galois over $\Q_{11}.$ | |
Intermediate fields
| $\Q_{11}(\sqrt{11})$, 11.7.6.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{11}$ |
| Relative Eisenstein polynomial: | \( x^{14} - 11 \) |
Invariants of the Galois closure
| Galois group: | $C_2\times C_7:C_3$ (as 14T5) |
| Inertia group: | $C_{14}$ |
| Unramified degree: | $3$ |
| Tame degree: | $14$ |
| Wild slopes: | None |
| Galois mean slope: | $13/14$ |
| Galois splitting model: | Not computed |