Defining polynomial
|
\(x^{15} + 31\)
|
Invariants
| Base field: | $\Q_{31}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $15$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{31}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 31 }) }$: | $15$ |
| This field is Galois and abelian over $\Q_{31}.$ | |
| Visible slopes: | None |
Intermediate fields
| 31.3.2.1, 31.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{31}$ |
| Relative Eisenstein polynomial: |
\( x^{15} + 31 \)
|
Ramification polygon
| Residual polynomials: | $z^{14} + 15z^{13} + 12z^{12} + 21z^{11} + z^{10} + 27z^{9} + 14z^{8} + 18z^{7} + 18z^{6} + 14z^{5} + 27z^{4} + z^{3} + 21z^{2} + 12z + 15$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_{15}$ (as 15T1) |
| Inertia group: | $C_{15}$ (as 15T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $1$ |
| Tame degree: | $15$ |
| Wild slopes: | None |
| Galois mean slope: | $14/15$ |
| Galois splitting model: | Not computed |