Defining polynomial
|
\(x^{8} + 7 x^{4} + 7 x^{3} + x^{2} + 7 x + 2\)
|
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $8$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{11}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 11 }) }$: | $8$ |
| This field is Galois and abelian over $\Q_{11}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{11}(\sqrt{2})$, 11.4.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 11.8.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of
\( x^{8} + 7 x^{4} + 7 x^{3} + x^{2} + 7 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x - 11 \)
$\ \in\Q_{11}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.