Defining polynomial
|
\(x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3\)
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $6$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $6$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 7 }) }$: | $6$ |
| This field is Galois and abelian over $\Q_{7}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{7}(\sqrt{3})$, 7.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 7.6.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of
\( x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3 \)
|
| 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 group: | $C_6$ (as 6T1) |
| Inertia group: | trivial |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $6$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: | $x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1$ |