Defining polynomial
|
$( x^{2} + 82 x + 3 )^{4} + 89$
|
Invariants
| Base field: | $\Q_{89}$ |
| Degree $d$: | $8$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{89}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{89})$ $=$$\Gal(K/\Q_{89})$: | $C_2\times C_4$ |
| This field is Galois and abelian over $\Q_{89}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $7920 = (89^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{89}(\sqrt{3})$, $\Q_{89}(\sqrt{89})$, $\Q_{89}(\sqrt{89\cdot 3})$, 89.2.2.2a1.2, 89.1.4.3a1.1, 89.1.4.3a1.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{89}(\sqrt{3})$ $\cong \Q_{89}(t)$ where $t$ is a root of
\( x^{2} + 82 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + 89 \)
$\ \in\Q_{89}(t)[x]$
|
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: | $C_2\times C_4$ (as 8T2) |
| 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: | not computed |