Defining polynomial
\(x^{8} - 165 x^{4} - 4356\) |
Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{11}(\sqrt{2})$ |
Root number: | $-1$ |
$\card{ \Aut(K/\Q_{ 11 }) }$: | $4$ |
This field is not Galois over $\Q_{11}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{11}(\sqrt{2})$, 11.4.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{11}(\sqrt{2})$ $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{2} + 7 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{4} + 33 t + 33 \) $\ \in\Q_{11}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{3} + 4z^{2} + 6z + 4$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $\OD_{16}$ (as 8T7) |
Inertia group: | Intransitive group isomorphic to $C_4$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $4$ |
Tame degree: | $4$ |
Wild slopes: | None |
Galois mean slope: | $3/4$ |
Galois splitting model: | $x^{8} - 4 x^{7} + 7 x^{6} - 7 x^{5} - 371 x^{4} + 749 x^{3} - 366 x^{2} - 9 x + 81$ |