Defining polynomial
|
\(x^{8} + 2 x^{5} + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $8$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $C_1$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\frac{12}{7}, \frac{12}{7}, \frac{12}{7}]$ |
| Visible Swan slopes: | $[\frac{5}{7},\frac{5}{7},\frac{5}{7}]$ |
| Means: | $\langle\frac{5}{14}, \frac{15}{28}, \frac{5}{8}\rangle$ |
| Rams: | $(\frac{5}{7}, \frac{5}{7}, \frac{5}{7})$ |
| Jump set: | $[1, 3, 6, 13]$ |
| Roots of unity: | $2$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 2 }$. |
Canonical tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{8} + 2 x^{5} + 2 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[5, 5, 5, 0]$ |
Invariants of the Galois closure
| Galois degree: | $168$ |
| Galois group: | $F_8:C_3$ (as 8T36) |
| Inertia group: | $F_8$ (as 8T25) |
| Wild inertia group: | $C_2^3$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $7$ |
| Galois Artin slopes: | $[\frac{12}{7}, \frac{12}{7}, \frac{12}{7}]$ |
| Galois Swan slopes: | $[\frac{5}{7},\frac{5}{7},\frac{5}{7}]$ |
| Galois mean slope: | $1.6071428571428572$ |
| Galois splitting model: | $x^{8} - 4 x^{7} + 14 x^{6} - 14 x^{5} + 14 x^{4} + 28 x^{3} + 14 x^{2} - 18 x + 23$ |