Defining polynomial
|
\(x^{9} + 18 x^{5} + 21\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $9$ |
| Ramification exponent $e$: | $9$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $22$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[5/2, 17/6]$ |
Intermediate fields
| 3.3.5.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}$ |
| Relative Eisenstein polynomial: |
\( x^{9} + 18 x^{5} + 21 \)
|
Ramification polygon
| Residual polynomials: | $2z + 1$,$z^{3} + 2$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[14, 9, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_3^2:C_6$ (as 9T11) |
| Inertia group: | $C_3^2:C_6$ (as 9T11) |
| Wild inertia group: | $\He_3$ |
| Unramified degree: | $1$ |
| Tame degree: | $2$ |
| Wild slopes: | $[2, 5/2, 17/6]$ |
| Galois mean slope: | $47/18$ |
| Galois splitting model: | $x^{9} - 6 x^{6} + 201 x^{3} - 512$ |