Defining polynomial
|
\(x^{9} + 5\)
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $9$ |
| Ramification exponent $e$: | $9$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{5}$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 5 }) }$: | $1$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible slopes: | None |
Intermediate fields
| 5.3.2.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{5}$ |
| Relative Eisenstein polynomial: |
\( x^{9} + 5 \)
|
Ramification polygon
| Residual polynomials: | $z^{8} + 4z^{7} + z^{6} + 4z^{5} + z^{4} + z^{3} + 4z^{2} + z + 4$ |
| Associated inertia: | $6$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_9:C_6$ (as 9T10) |
| Inertia group: | $C_9$ (as 9T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $6$ |
| Tame degree: | $9$ |
| Wild slopes: | None |
| Galois mean slope: | $8/9$ |
| Galois splitting model: | $x^{9} - 5$ |