Defining polynomial
|
\(x^{22} + 218\)
|
Invariants
| Base field: | $\Q_{109}$ |
| Degree $d$: | $22$ |
| Ramification exponent $e$: | $22$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $21$ |
| Discriminant root field: | $\Q_{109}(\sqrt{109\cdot 2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 109 }) }$: | $2$ |
| This field is not Galois over $\Q_{109}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{109}(\sqrt{109\cdot 2})$, 109.11.10.1 |
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: |
\( x^{22} + 218 \)
|
Ramification polygon
| Residual polynomials: | $z^{21} + 22z^{20} + 13z^{19} + 14z^{18} + 12z^{17} + 65z^{16} + 57z^{15} + 68z^{14} + 73z^{13} + 53z^{12} + 58z^{11} + 93z^{10} + 58z^{9} + 53z^{8} + 73z^{7} + 68z^{6} + 57z^{5} + 65z^{4} + 12z^{3} + 14z^{2} + 13z + 22$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $D_{22}$ (as 22T3) |
| Inertia group: | $C_{22}$ (as 22T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $22$ |
| Wild slopes: | None |
| Galois mean slope: | $21/22$ |
| Galois splitting model: | Not computed |