Defining polynomial
\(x^{9} - 25 x^{3} + 250\)  |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $9$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 5 })|$: | $3$ |
| This field is not Galois over $\Q_{5}.$ | |
Intermediate fields
| 5.3.2.1, 5.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 5.3.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{3} - x + 2 \) |
| Relative Eisenstein polynomial: | \( x^{3} - 5 t \)$\ \in\Q_{5}(t)[x]$ |
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: | $x^{9} - 4 x^{8} + 3 x^{7} + 6 x^{6} - 18 x^{5} + 26 x^{4} - 8 x^{3} - 15 x^{2} + 14 x - 13$ |