Defining polynomial
|
$( x^{2} + 192 x + 2 )^{4} + 197$
|
Invariants
| Base field: | $\Q_{197}$ |
| Degree $d$: | $8$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{197}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{197})$ $=$ $\Gal(K/\Q_{197})$: | $C_2\times C_4$ |
| This field is Galois and abelian over $\Q_{197}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $38808 = (197^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{197}(\sqrt{2})$, $\Q_{197}(\sqrt{197})$, $\Q_{197}(\sqrt{197\cdot 2})$, 197.2.2.2a1.2, 197.1.4.3a1.3, 197.1.4.3a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{197}(\sqrt{2})$ $\cong \Q_{197}(t)$ where $t$ is a root of
\( x^{2} + 192 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + 197 \)
$\ \in\Q_{197}(t)[x]$
|
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 |