Defining polynomial
\(x^{9} - 11\)  |
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $9$ |
| Ramification exponent $e$: | $9$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{11}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 11 })|$: | $1$ |
| This field is not Galois over $\Q_{11}.$ | |
Intermediate fields
| 11.3.2.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{11}$ |
| Relative Eisenstein polynomial: | \( x^{9} - 11 \) |
Invariants of the Galois closure
| Galois group: | $D_9:C_3$ (as 9T10) |
| Inertia group: | $C_9$ |
| Unramified degree: | $6$ |
| Tame degree: | $9$ |
| Wild slopes: | None |
| Galois mean slope: | $8/9$ |
| Galois splitting model: | $x^{9} - 11$ |