Defining polynomial
|
\(x^{10} + 21\)
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $10$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{7}(\sqrt{7})$ |
| Root number: | $i$ |
| $\card{ \Aut(K/\Q_{ 7 }) }$: | $2$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{7}(\sqrt{7})$, 7.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: |
\( x^{10} + 21 \)
|
Ramification polygon
| Residual polynomials: | $z^{9} + 3z^{8} + 3z^{7} + z^{6} + z^{2} + 3z + 3$ |
| Associated inertia: | $4$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times F_5$ (as 10T5) |
| Inertia group: | $C_{10}$ (as 10T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $4$ |
| Tame degree: | $10$ |
| Wild slopes: | None |
| Galois mean slope: | $9/10$ |
| Galois splitting model: | $x^{10} - 7$ |