Defining polynomial
| \( x^{10} + 15 x^{6} + 10 x^{5} + 100 x^{2} + 75 x + 25 \) |
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: | $-1$ |
| $|\Aut(K/\Q_{ 5 })|$: | $1$ |
| This field is not Galois over $\Q_{5}.$ | |
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 \( x^{2} + 2 \) |
| Relative Eisenstein polynomial: | $ x^{5} + \left(10 t + 20\right) x + 5 \in\Q_{5}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $D_5:F_5$ (as 10T17) |
| Inertia group: | Intransitive group isomorphic to $C_5^2:C_4$ |
| Unramified degree: | $2$ |
| Tame degree: | $4$ |
| Wild slopes: | [5/4, 5/4] |
| Galois mean slope: | $123/100$ |
| Galois splitting model: | $x^{10} - 10 x^{8} + 35 x^{6} - 4 x^{5} - 50 x^{4} + 20 x^{3} + 25 x^{2} - 20 x - 28$ |