Defining polynomial
\(x^{4} + 8 x^{2} + 10 x + 2\)
|
Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $4$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{11}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 11 }) }$: | $4$ |
This field is Galois and abelian over $\Q_{11}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{11}(\sqrt{2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 11.4.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of
\( x^{4} + 8 x^{2} + 10 x + 2 \)
|
Relative Eisenstein polynomial: |
\( x - 11 \)
$\ \in\Q_{11}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
Galois group: | $C_4$ (as 4T1) |
Inertia group: | trivial |
Wild inertia group: | $C_1$ |
Unramified degree: | $4$ |
Tame degree: | $1$ |
Wild slopes: | None |
Galois mean slope: | $0$ |
Galois splitting model: | $x^{4} - 4 x^{2} + 2$ |