Defining polynomial
|
\(x^{9} + 3 x^{5} + 6 x^{3} + 3\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $9$ |
| Ramification exponent $e$: | $9$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $13$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3\cdot 2})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[3/2, 5/3]$ |
Intermediate fields
| 3.3.3.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}$ |
| Relative Eisenstein polynomial: |
\( x^{9} + 3 x^{5} + 6 x^{3} + 3 \)
|
Ramification polygon
| Residual polynomials: | $z^{2} + 1$,$z^{3} + 1$ |
| Associated inertia: | $2$,$1$ |
| Indices of inseparability: | $[5, 3, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_3^2:D_6$ (as 9T18) |
| Inertia group: | $C_3^2:S_3$ (as 9T12) |
| Wild inertia group: | $\He_3$ |
| Unramified degree: | $2$ |
| Tame degree: | $2$ |
| Wild slopes: | $[3/2, 3/2, 5/3]$ |
| Galois mean slope: | $85/54$ |
| Galois splitting model: | $x^{9} - 3 x^{7} - 3 x^{6} - 9 x^{5} - 18 x^{4} + 27 x^{2} + 27 x + 9$ |