Defining polynomial
|
\(x^{9} + 14\)
|
Invariants
| Base field: | $\Q_{7}$ |
|
| Degree $d$: | $9$ |
|
| Ramification index $e$: | $9$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $8$ |
|
| Discriminant root field: | $\Q_{7}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{7})$: | $C_3$ | |
| This field is not Galois over $\Q_{7}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $6 = (7 - 1)$ |
|
Intermediate fields
| $\Q_{7}(\sqrt[3]{7 \cdot 2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{7}$ |
|
| Relative Eisenstein polynomial: |
\( x^{9} + 14 \)
|
Ramification polygon
| Residual polynomials: | $z^8 + 2 z^7 + z^6 + z + 2$ |
| Associated inertia: | $3$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $27$ |
| Galois group: | $C_9:C_3$ (as 9T6) |
| Inertia group: | $C_9$ (as 9T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $9$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.8888888888888888$ |
| Galois splitting model: | $x^{9} - 63 x^{7} + 1323 x^{5} - 10290 x^{3} + 21609 x - 12005$ |