Defining polynomial
|
\(x^{14} + 29\)
|
Invariants
| Base field: | $\Q_{29}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $13$ |
| Discriminant root field: | $\Q_{29}(\sqrt{29})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 29 }) }$: | $14$ |
| This field is Galois and abelian over $\Q_{29}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{29}(\sqrt{29})$, 29.7.6.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{29}$ |
| Relative Eisenstein polynomial: |
\( x^{14} + 29 \)
|
Ramification polygon
| Residual polynomials: | $z^{13} + 14z^{12} + 4z^{11} + 16z^{10} + 15z^{9} + z^{8} + 16z^{7} + 10z^{6} + 16z^{5} + z^{4} + 15z^{3} + 16z^{2} + 4z + 14$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_{14}$ (as 14T1) |
| Inertia group: | $C_{14}$ (as 14T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $1$ |
| Tame degree: | $14$ |
| Wild slopes: | None |
| Galois mean slope: | $13/14$ |
| Galois splitting model: | Not computed |