Defining polynomial
|
\(x^{8} + 137\)
|
Invariants
| Base field: | $\Q_{137}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $8$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $7$ |
| Discriminant root field: | $\Q_{137}(\sqrt{137})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 137 }) }$: | $8$ |
| This field is Galois and abelian over $\Q_{137}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{137}(\sqrt{137})$, 137.4.3.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{137}$ |
| Relative Eisenstein polynomial: |
\( x^{8} + 137 \)
|
Ramification polygon
| Residual polynomials: | $z^{7} + 8z^{6} + 28z^{5} + 56z^{4} + 70z^{3} + 56z^{2} + 28z + 8$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_8$ (as 8T1) |
| Inertia group: | $C_8$ (as 8T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $1$ |
| Tame degree: | $8$ |
| Wild slopes: | None |
| Galois mean slope: | $7/8$ |
| Galois splitting model: | Not computed |