Defining polynomial
|
\(x^{22} + 109\)
|
Invariants
| Base field: | $\Q_{109}$ |
|
| Degree $d$: | $22$ |
|
| Ramification index $e$: | $22$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $21$ |
|
| Discriminant root field: | $\Q_{109}(\sqrt{109})$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{109})$: | $C_2$ | |
| This field is not Galois over $\Q_{109}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $108 = (109 - 1)$ |
|
Intermediate fields
| $\Q_{109}(\sqrt{109})$, 109.1.11.10a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{109}$ |
|
| Relative Eisenstein polynomial: |
\( x^{22} + 109 \)
|
Ramification polygon
| Residual polynomials: | $z^{21} + 22 z^{20} + 13 z^{19} + 14 z^{18} + 12 z^{17} + 65 z^{16} + 57 z^{15} + 68 z^{14} + 73 z^{13} + 53 z^{12} + 58 z^{11} + 93 z^{10} + 58 z^9 + 53 z^8 + 73 z^7 + 68 z^6 + 57 z^5 + 65 z^4 + 12 z^3 + 14 z^2 + 13 z + 22$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $44$ |
| Galois group: | $D_{22}$ (as 22T3) |
| Inertia group: | $C_{22}$ (as 22T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $22$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.9545454545454546$ |
| Galois splitting model: | not computed |