Defining polynomial
| \( x^{9} - 31 \) |
Invariants
| Base field: | $\Q_{31}$ |
| Degree $d$ : | $9$ |
| Ramification exponent $e$ : | $9$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $8$ |
| Discriminant root field: | $\Q_{31}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 31 })|$: | $3$ |
| This field is not Galois over $\Q_{31}$. | |
Intermediate fields
| 31.3.2.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{31}$ |
| Relative Eisenstein polynomial: | \( x^{9} - 31 \) |
Invariants of the Galois closure
| Galois group: | $C_9:C_3$ (as 9T6) |
| Inertia group: | $C_9$ |
| Unramified degree: | $3$ |
| Tame degree: | $9$ |
| Wild slopes: | None |
| Galois mean slope: | $8/9$ |
| Galois splitting model: | Not computed |