Defining polynomial
|
\(x^{9} + 6 x^{8} + 3\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $9$ |
| Ramification index $e$: | $9$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $16$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{3})$: | $C_3$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible Artin slopes: | $[2, 2]$ |
| Visible Swan slopes: | $[1,1]$ |
| Means: | $\langle\frac{2}{3}, \frac{8}{9}\rangle$ |
| Rams: | $(1, 1)$ |
| Jump set: | undefined |
| Roots of unity: | $2 = (3 - 1)$ |
Intermediate fields
| 3.1.3.4a2.1, 3.1.3.4a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{3}$ |
| Relative Eisenstein polynomial: |
\( x^{9} + 6 x^{8} + 3 \)
|
Ramification polygon
| Residual polynomials: | $z^8 + 2$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[8, 8, 0]$ |
Invariants of the Galois closure
| Galois degree: | $18$ |
| Galois group: | $C_3\times S_3$ (as 9T4) |
| Inertia group: | $C_3^2$ (as 9T2) |
| Wild inertia group: | $C_3^2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2]$ |
| Galois Swan slopes: | $[1,1]$ |
| Galois mean slope: | $1.7777777777777777$ |
| Galois splitting model: | $x^{9} - 3 x^{8} + 3 x^{7} - 6 x^{5} + 15 x^{4} - 21 x^{3} + 15 x^{2} - 6 x + 1$ |