Defining polynomial
\(x^{8} + 2 x^{7} + 2 x^{2} + 4 x + 2\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $8$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $14$ |
Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[4/3, 4/3, 5/2]$ |
Intermediate fields
2.4.4.5 |
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: | \( x^{8} + 2 x^{7} + 2 x^{2} + 4 x + 2 \) |
Ramification polygon
Residual polynomials: | $z + 1$,$z^{2} + 1$ |
Associated inertia: | $1$,$1$ |
Indices of inseparability: | $[7, 2, 2, 0]$ |
Invariants of the Galois closure
Galois group: | $C_2\wr S_4$ (as 8T44) |
Inertia group: | $C_2\wr A_4$ (as 8T38) |
Wild inertia group: | $C_2\wr C_2^2$ |
Unramified degree: | $2$ |
Tame degree: | $3$ |
Wild slopes: | $[4/3, 4/3, 2, 7/3, 7/3, 5/2]$ |
Galois mean slope: | $223/96$ |
Galois splitting model: | $x^{8} + 4 x^{2} + 28$ |