Defining polynomial
|
\(x^{8} + x^{4} + 12 x^{3} + 10 x^{2} + 3 x + 2\)
|
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $8$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{19}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 19 }) }$: | $8$ |
| This field is Galois and abelian over $\Q_{19}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{19}(\sqrt{2})$, 19.4.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 19.8.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of
\( x^{8} + x^{4} + 12 x^{3} + 10 x^{2} + 3 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x - 19 \)
$\ \in\Q_{19}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.