Defining polynomial
| \( x^{10} + 15 x^{3} + 10 \) |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$ : | $10$ |
| Ramification exponent $e$ : | $10$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $12$ |
| Discriminant root field: | $\Q_{5}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 5 })|$: | $1$ |
| This field is not Galois over $\Q_{5}$. | |
Intermediate fields
| $\Q_{5}(\sqrt{5*})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{5}$ |
| Relative Eisenstein polynomial: | \( x^{10} + 40 x^{6} + 40 x^{3} + 10 \) |
Invariants of the Galois closure
| Galois group: | $D_5^2.C_4$ (as 10T28) |
| Inertia group: | $C_5^2 : C_8$ |
| Unramified degree: | $2$ |
| Tame degree: | $8$ |
| Wild slopes: | [11/8, 11/8] |
| Galois mean slope: | $271/200$ |
| Galois splitting model: | $x^{10} - 20 x^{8} - 40 x^{7} + 70 x^{6} + 472 x^{5} + 1140 x^{4} + 1960 x^{3} - 3615 x^{2} - 25560 x - 36504$ |