Defining polynomial
|
\(x^{13} + 9 x + 18\)
|
Invariants
| Base field: | $\Q_{23}$ |
| Degree $d$: | $13$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $13$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{23}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 23 }) }$: | $13$ |
| This field is Galois and abelian over $\Q_{23}.$ | |
| Visible slopes: | None |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 23 }$. |
Unramified/totally ramified tower
| Unramified subfield: | 23.13.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of
\( x^{13} + 9 x + 18 \)
|
| Relative Eisenstein polynomial: |
\( x - 23 \)
$\ \in\Q_{23}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.