Defining polynomial
|
\(x^{17} + 3 x + 35\)
|
Invariants
| Base field: | $\Q_{37}$ |
| Degree $d$: | $17$ |
| Ramification index $e$: | $1$ |
| Residue field degree $f$: | $17$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{37}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{37})$ $=$$\Gal(K/\Q_{37})$: | $C_{17}$ |
| This field is Galois and abelian over $\Q_{37}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $456487940826035155404146916 = (37^{ 17 } - 1)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 37 }$. |
Canonical tower
| Unramified subfield: | 37.17.1.0a1.1 $\cong \Q_{37}(t)$ where $t$ is a root of
\( x^{17} + 3 x + 35 \)
|
| Relative Eisenstein polynomial: |
\( x - 37 \)
$\ \in\Q_{37}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
| Galois degree: | $17$ |
| Galois group: | $C_{17}$ (as 17T1) |
| Inertia group: | trivial |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $17$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.0$ |
| Galois splitting model: | not computed |