Defining polynomial
|
\(x^{11} + 3 x + 3\)
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $11$ |
| Ramification index $e$: | $1$ |
| Residue field degree $f$: | $11$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{5}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{5})$ $=$$\Gal(K/\Q_{5})$: | $C_{11}$ |
| This field is Galois and abelian over $\Q_{5}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $48828124 = (5^{ 11 } - 1)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 5 }$. |
Canonical tower
| Unramified subfield: | 5.11.1.0a1.1 $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{11} + 3 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x - 5 \)
$\ \in\Q_{5}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.