Defining polynomial
\(x^{8} + 8 x^{6} + 20 x^{4} + 34\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $8$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $31$ |
Discriminant root field: | $\Q_{2}(\sqrt{2})$ |
Root number: | $-1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $4$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[3, 4, 5]$ |
Intermediate fields
$\Q_{2}(\sqrt{2})$, 2.4.11.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{2}$ |
Relative Eisenstein polynomial: | \( x^{8} + 8 x^{6} + 20 x^{4} + 34 \) |
Ramification polygon
Residual polynomials: | $z + 1$,$z^{2} + 1$,$z^{4} + 1$ |
Associated inertia: | $1$,$1$,$1$ |
Indices of inseparability: | $[24, 16, 8, 0]$ |
Invariants of the Galois closure
Galois group: | $\OD_{16}$ (as 8T7) |
Inertia group: | $\OD_{16}$ (as 8T7) |
Wild inertia group: | $\OD_{16}$ |
Unramified degree: | $1$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 3, 4, 5]$ |
Galois mean slope: | $4$ |
Galois splitting model: | $x^{8} - 24 x^{6} - 108 x^{4} + 162$ |