Defining polynomial
|
$( x^{2} + 6 x + 3 )^{4} + 7$
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $8$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{7}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{7})$ $=$ $\Gal(K/\Q_{7})$: | $D_4$ |
| This field is Galois over $\Q_{7}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $48 = (7^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{7}(\sqrt{3})$, $\Q_{7}(\sqrt{7})$, $\Q_{7}(\sqrt{7\cdot 3})$, 7.2.2.2a1.2, 7.1.4.3a1.1 x2, 7.1.4.3a1.2 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of
\( x^{2} + 6 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + 7 \)
$\ \in\Q_{7}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^3 + 4 z^2 + 6 z + 4$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $8$ |
| Galois group: | $D_4$ (as 8T4) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.75$ |
| Galois splitting model: | $x^{8} - 2 x^{7} + 2 x^{6} + 4 x^{5} - x^{4} + 4 x^{3} + 2 x^{2} - 2 x + 1$ |