Defining polynomial
|
\(x^{9} + 109\)
|
Invariants
| Base field: | $\Q_{109}$ |
| Degree $d$: | $9$ |
| Ramification exponent $e$: | $9$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{109}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 109 }) }$: | $9$ |
| This field is Galois and abelian over $\Q_{109}.$ | |
| Visible slopes: | None |
Intermediate fields
| 109.3.2.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{109}$ |
| Relative Eisenstein polynomial: |
\( x^{9} + 109 \)
|
Ramification polygon
| Residual polynomials: | $z^{8} + 9z^{7} + 36z^{6} + 84z^{5} + 17z^{4} + 17z^{3} + 84z^{2} + 36z + 9$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_9$ (as 9T1) |
| Inertia group: | $C_9$ (as 9T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $1$ |
| Tame degree: | $9$ |
| Wild slopes: | None |
| Galois mean slope: | $8/9$ |
| Galois splitting model: | Not computed |