Defining polynomial
|
\(x^{8} + 4 x^{7} + 2 x^{4} + 4 x^{2} + 6\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $8$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $22$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$ $=$$\Gal(K/\Q_{2})$: | $D_4$ |
| This field is Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2, 3, \frac{7}{2}]$ |
| Visible Swan slopes: | $[1,2,\frac{5}{2}]$ |
| Means: | $\langle\frac{1}{2}, \frac{5}{4}, \frac{15}{8}\rangle$ |
| Rams: | $(1, 3, 5)$ |
| Jump set: | $[1, 2, 4, 16]$ |
| Roots of unity: | $2$ |
Intermediate fields
| $\Q_{2}(\sqrt{-5})$, $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{2\cdot 5})$, 2.1.4.8b1.5, 2.1.4.9a1.8 x2, 2.1.4.10a1.5 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{8} + 4 x^{7} + 2 x^{4} + 4 x^{2} + 6 \)
|
Ramification polygon
| Residual polynomials: | $z^4 + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$,$1$ |
| Indices of inseparability: | $[15, 10, 4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $8$ |
| Galois group: | $D_4$ (as 8T4) |
| Inertia group: | $D_4$ (as 8T4) |
| Wild inertia group: | $D_4$ |
| Galois unramified degree: | $1$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 3, \frac{7}{2}]$ |
| Galois Swan slopes: | $[1,2,\frac{5}{2}]$ |
| Galois mean slope: | $2.75$ |
| Galois splitting model: | $x^{8} - 4 x^{7} + 4 x^{6} - 4 x^{5} + 12 x^{4} + 4 x^{3} + 4 x^{2} + 4 x + 1$ |