Defining polynomial
\(x^{8} + 8 x^{6} - 4 x^{5} + 28 x^{4} + 8 x^{3} + 88 x^{2} + 72 x + 108\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $16$ |
Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $4$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[2, 3]$ |
Intermediate fields
$\Q_{2}(\sqrt{5})$, 2.4.4.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} + x + 1 \) |
Relative Eisenstein polynomial: | \( x^{4} + \left(4 t + 6\right) x^{2} + 4 t x + 12 t + 6 \) $\ \in\Q_{2}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + t$,$z^{2} + 1$ |
Associated inertia: | $1$,$1$ |
Indices of inseparability: | $[5, 2, 0]$ |
Invariants of the Galois closure
Galois group: | $\OD_{16}$ (as 8T7) |
Inertia group: | Intransitive group isomorphic to $C_2\times C_4$ |
Wild inertia group: | $C_2\times C_4$ |
Unramified degree: | $2$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 3, 3]$ |
Galois mean slope: | $5/2$ |
Galois splitting model: | $x^{8} + 10 x^{6} + 25 x^{4} + 20 x^{2} + 5$ |