Defining polynomial
| \( x^{14} - 127 \) |
Invariants
| Base field: | $\Q_{127}$ |
| Degree $d$ : | $14$ |
| Ramification exponent $e$ : | $14$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $13$ |
| Discriminant root field: | $\Q_{127}(\sqrt{127})$ |
| Root number: | $i$ |
| $|\Gal(K/\Q_{ 127 })|$: | $14$ |
| This field is Galois and abelian over $\Q_{127}$. | |
Intermediate fields
| $\Q_{127}(\sqrt{127})$, 127.7.6.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{127}$ |
| Relative Eisenstein polynomial: | \( x^{14} - 127 \) |
Invariants of the Galois closure
| Galois group: | $C_{14}$ (as 14T1) |
| Inertia group: | $C_{14}$ |
| Unramified degree: | $1$ |
| Tame degree: | $14$ |
| Wild slopes: | None |
| Galois mean slope: | $13/14$ |
| Galois splitting model: | Not computed |