Defining polynomial
\(x^{4} + 8 x^{3} + 8 x + 2\)
|
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $11$ |
Discriminant root field: | $\Q_{2}(\sqrt{-2\cdot 5})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[3, 4]$ |
Intermediate fields
$\Q_{2}(\sqrt{-2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{2}$ |
Relative Eisenstein polynomial: |
\( x^{4} + 8 x^{3} + 8 x + 2 \)
|
Ramification polygon
Residual polynomials: | $z + 1$,$z^{2} + 1$ |
Associated inertia: | $1$,$1$ |
Indices of inseparability: | $[8, 4, 0]$ |
Invariants of the Galois closure
Galois group: | $D_4$ (as 4T3) |
Inertia group: | $C_4$ (as 4T1) |
Wild inertia group: | $C_4$ |
Unramified degree: | $2$ |
Tame degree: | $1$ |
Wild slopes: | $[3, 4]$ |
Galois mean slope: | $11/4$ |
Galois splitting model: | $x^{4} + 4 x^{2} + 6$ |