Defining polynomial
|
\(x^{9} + 711\)
|
Invariants
| Base field: | $\Q_{79}$ |
|
| Degree $d$: | $9$ |
|
| Ramification index $e$: | $9$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $8$ |
|
| Discriminant root field: | $\Q_{79}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{79})$: | $C_3$ | |
| This field is not Galois over $\Q_{79}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $78 = (79 - 1)$ |
|
Intermediate fields
| 79.1.3.2a1.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{79}$ |
|
| Relative Eisenstein polynomial: |
\( x^{9} + 711 \)
|
Ramification polygon
| Residual polynomials: | $z^8 + 9 z^7 + 36 z^6 + 5 z^5 + 47 z^4 + 47 z^3 + 5 z^2 + 36 z + 9$ |
| 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: | not computed |