Defining polynomial
|
\(x^{21} + 43\)
|
Invariants
| Base field: | $\Q_{43}$ |
| Degree $d$: | $21$ |
| Ramification exponent $e$: | $21$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $20$ |
| Discriminant root field: | $\Q_{43}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 43 }) }$: | $21$ |
| This field is Galois and abelian over $\Q_{43}.$ | |
| Visible slopes: | None |
Intermediate fields
| 43.3.2.1, 43.7.6.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{43}$ |
| Relative Eisenstein polynomial: |
\( x^{21} + 43 \)
|
Ramification polygon
| Residual polynomials: | $z^{20} + 21z^{19} + 38z^{18} + 40z^{17} + 8z^{16} + 10z^{15} + 41z^{14} + 8z^{13} + 14z^{12} + 25z^{11} + 30z^{10} + 30z^{9} + 25z^{8} + 14z^{7} + 8z^{6} + 41z^{5} + 10z^{4} + 8z^{3} + 40z^{2} + 38z + 21$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_{21}$ (as 21T1) |
| Inertia group: | $C_{21}$ (as 21T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $1$ |
| Tame degree: | $21$ |
| Wild slopes: | None |
| Galois mean slope: | $20/21$ |
| Galois splitting model: | Not computed |