Defining polynomial
| \( x^{9} - 4 x^{3} + 8 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $9$ |
| Ramification exponent $e$ : | $3$ |
| Residue field degree $f$ : | $3$ |
| Discriminant exponent $c$ : | $6$ |
| Discriminant root field: | $\Q_{2}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $3$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.3.2.1, 2.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{3} - x + 1 \) |
| Relative Eisenstein polynomial: | $ x^{3} - 2 t \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times S_3$ (as 9T4) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Unramified degree: | $6$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: | $x^{9} - 3 x^{7} - x^{6} + 3 x^{5} + 2 x^{4} + 4 x^{3} - x^{2} - 5 x + 1$ |