Defining polynomial
| \( x^{10} + 10 x^{8} + 10 x^{7} + 15 x^{6} - 10 x^{5} + 5 x^{4} + 5 x^{2} - 15 x + 7 \) |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$ : | $10$ |
| Ramification exponent $e$ : | $5$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $12$ |
| Discriminant root field: | $\Q_{5}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 5 })|$: | $2$ |
| This field is not Galois over $\Q_{5}$. | |
Intermediate fields
| $\Q_{5}(\sqrt{*})$, 5.5.6.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{5}(\sqrt{*})$ $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{2} + 2 \) |
| Relative Eisenstein polynomial: | $ x^{5} + 5 x^{2} + 5 \in\Q_{5}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $F_5$ (as 10T4) |
| Inertia group: | Intransitive group isomorphic to $D_5$ |
| Unramified degree: | $2$ |
| Tame degree: | $2$ |
| Wild slopes: | [3/2] |
| Galois mean slope: | $13/10$ |
| Galois splitting model: | $x^{10} + 10 x^{8} - 20 x^{7} - 45 x^{6} + 20 x^{5} - 500 x^{4} - 120 x^{3} + 1490 x^{2} - 760 x + 248$ |