Defining polynomial
| \( x^{10} + 20 x + 10 \) |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$ : | $10$ |
| Ramification exponent $e$ : | $10$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $10$ |
| 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: | \( x^{10} + 10 x^{2} + 20 x + 10 \) |
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: | [9/8, 9/8] |
| Galois mean slope: | $223/200$ |
| Galois splitting model: | $x^{10} - 1955 x^{8} - 580 x^{7} - 2001110 x^{6} + 7057964 x^{5} - 243542310 x^{4} + 3251676820 x^{3} + 49946545205 x^{2} + 101082545540 x - 111230164631$ |