Defining polynomial
|
\(x^{17} + 12 x + 51\)
|
Invariants
| Base field: | $\Q_{53}$ |
| Degree $d$: | $17$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $17$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{53}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 53 }) }$: | $17$ |
| This field is Galois and abelian over $\Q_{53}.$ | |
| Visible slopes: | None |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 53 }$. |
Unramified/totally ramified tower
| Unramified subfield: | 53.17.0.1 $\cong \Q_{53}(t)$ where $t$ is a root of
\( x^{17} + 12 x + 51 \)
|
| Relative Eisenstein polynomial: |
\( x - 53 \)
$\ \in\Q_{53}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
| Galois group: | $C_{17}$ (as 17T1) |
| Inertia group: | trivial |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $17$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: | Not computed |