Defining polynomial
|
\(x^{5} + x + 4\)
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $5$ |
| Ramification index $e$: | $1$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{7}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{7})$ $=$$\Gal(K/\Q_{7})$: | $C_5$ |
| This field is Galois and abelian over $\Q_{7}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $16806 = (7^{ 5 } - 1)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$. |
Canonical tower
| Unramified subfield: | 7.5.1.0a1.1 $\cong \Q_{7}(t)$ where $t$ is a root of
\( x^{5} + x + 4 \)
|
| Relative Eisenstein polynomial: |
\( x - 7 \)
$\ \in\Q_{7}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
| Galois degree: | $5$ |
| Galois group: | $C_5$ (as 5T1) |
| Inertia group: | trivial |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $5$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.0$ |
| Galois splitting model: | $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ |