Defining polynomial
|
\(x^{8} - 4 x^{6} + 8 x^{5} + 20 x^{4} - 16 x^{3} + 8 x^{2} + 16 x + 4\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $16$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $1$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[8/3, 8/3]$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{2} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + 4 t x^{2} + 4 x + 2 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[5, 4, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2^2:S_4$ (as 8T34) |
| Inertia group: | Intransitive group isomorphic to $C_2^2:A_4$ |
| Wild inertia group: | $C_2^4$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | $[4/3, 4/3, 8/3, 8/3]$ |
| Galois mean slope: | $55/24$ |
| Galois splitting model: | $x^{8} + 2 x^{6} + 7 x^{4} - 88 x^{3} + 226 x^{2} - 220 x + 75$ |
Additional information
This is the only octic extension of $\Q_p$ for any prime $p$ with Galois group $C_2^2:S_4$. Based on general restrictions of the higher ramification filtration, the only prime $p$ for which there can be $S_4$ extensions is $p=2$, and the Galois closure of this field is the compositum of all such extensions.