Defining polynomial
|
\(x^{3} + x + 14\)
|
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $3$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{17}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 17 }) }$: | $3$ |
| This field is Galois and abelian over $\Q_{17}.$ | |
| Visible slopes: | None |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 17 }$. |
Unramified/totally ramified tower
| Unramified subfield: | 17.3.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of
\( x^{3} + x + 14 \)
|
| Relative Eisenstein polynomial: |
\( x - 17 \)
$\ \in\Q_{17}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
| Galois group: | $C_3$ (as 3T1) |
| Inertia group: | trivial |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $3$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: | $x^{3} - x^{2} - 2 x + 1$ |