Defining polynomial
|
\(x^{9} + 15 x^{3} + 162 x^{2} + 127 x + 161\)
|
Invariants
| Base field: | $\Q_{163}$ |
|
| Degree $d$: | $9$ |
|
| Ramification index $e$: | $1$ |
|
| Residue field degree $f$: | $9$ |
|
| Discriminant exponent $c$: | $0$ |
|
| Discriminant root field: | $\Q_{163}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{163})$ $=$ $\Gal(K/\Q_{163})$: | $C_9$ | |
| This field is Galois and abelian over $\Q_{163}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $81224760533853742722 = (163^{ 9 } - 1)$ |
|
Intermediate fields
| 163.3.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 163.9.1.0a1.1 $\cong \Q_{163}(t)$ where $t$ is a root of
\( x^{9} + 15 x^{3} + 162 x^{2} + 127 x + 161 \)
|
|
| Relative Eisenstein polynomial: |
\( x - 163 \)
$\ \in\Q_{163}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.