Defining polynomial
|
\(x^{8} + 100 x^{4} - 500 x^{2} + 1250\)
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 5 }) }$: | $8$ |
| This field is Galois and abelian over $\Q_{5}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{5}(\sqrt{2})$, 5.4.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 5.4.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{4} + 4 x^{2} + 4 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 5 t \)
$\ \in\Q_{5}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_8$ (as 8T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $4$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | $x^{8} - x^{7} - 24 x^{6} + 23 x^{5} + 151 x^{4} - 197 x^{3} - 214 x^{2} + 429 x - 169$ |