Defining polynomial
| \( x^{9} - x + 3 \) |
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$ : | $9$ |
| Ramification exponent $e$ : | $1$ |
| Residue field degree $f$ : | $9$ |
| Discriminant exponent $c$ : | $0$ |
| Discriminant root field: | $\Q_{11}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 11 })|$: | $9$ |
| This field is Galois and abelian over $\Q_{11}$. | |
Intermediate fields
| 11.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 11.9.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{9} - x + 3 \) |
| Relative Eisenstein polynomial: | $ x - 11 \in\Q_{11}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_9$ (as 9T1) |
| Inertia group: | Trivial |
| Unramified degree: | $9$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: | $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ |