Defining polynomial
|
\(x^{10} - 10 x^{6} + 10 x^{5} - 25 x^{2} - 50 x + 25\)
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $5$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 5 }) }$: | $1$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible slopes: | $[5/4]$ |
Intermediate fields
| $\Q_{5}(\sqrt{2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{2} + 4 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{5} + \left(5 t + 5\right) x + 5 \)
$\ \in\Q_{5}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 4t + 4$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $D_5:F_5$ (as 10T17) |
| Inertia group: | Intransitive group isomorphic to $C_5:F_5$ |
| Wild inertia group: | $C_5^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $4$ |
| Wild slopes: | $[5/4, 5/4]$ |
| Galois mean slope: | $123/100$ |
| Galois splitting model: | $x^{10} - 10 x^{8} + 35 x^{6} - 4 x^{5} - 50 x^{4} + 20 x^{3} + 25 x^{2} - 20 x - 28$ |