Defining polynomial
|
\(x^{12} + 23\)
|
Invariants
| Base field: | $\Q_{23}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $11$ |
| Discriminant root field: | $\Q_{23}(\sqrt{23})$ |
| Root number: | $i$ |
| $\card{ \Aut(K/\Q_{ 23 }) }$: | $2$ |
| This field is not Galois over $\Q_{23}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{23}(\sqrt{23\cdot 5})$, 23.3.2.1, 23.4.3.1, 23.6.5.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{23}$ |
| Relative Eisenstein polynomial: |
\( x^{12} + 23 \)
|
Ramification polygon
| Residual polynomials: | $z^{11} + 12z^{10} + 20z^{9} + 13z^{8} + 12z^{7} + 10z^{6} + 4z^{5} + 10z^{4} + 12z^{3} + 13z^{2} + 20z + 12$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $D_{12}$ (as 12T12) |
| Inertia group: | $C_{12}$ (as 12T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $12$ |
| Wild slopes: | None |
| Galois mean slope: | $11/12$ |
| Galois splitting model: | Not computed |