Defining polynomial
|
\(x^{19} + 18 x + 17\)
|
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $19$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $19$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{19}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 19 }) }$: | $19$ |
| This field is Galois and abelian over $\Q_{19}.$ | |
| Visible slopes: | None |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 19 }$. |
Unramified/totally ramified tower
| Unramified subfield: | 19.19.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of
\( x^{19} + 18 x + 17 \)
|
| Relative Eisenstein polynomial: |
\( x - 19 \)
$\ \in\Q_{19}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.