Defining polynomial
| \( x^{10} - 15 x^{6} + 5 \) |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$ : | $10$ |
| Ramification exponent $e$ : | $10$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $15$ |
| Discriminant root field: | $\Q_{5}(\sqrt{5})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 5 })|$: | $2$ |
| This field is not Galois over $\Q_{5}$. | |
Intermediate fields
| $\Q_{5}(\sqrt{5})$, 5.5.7.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{5}$ |
| Relative Eisenstein polynomial: | \( x^{10} - 40 x^{6} + 400 x^{2} + 5 \) |
Invariants of the Galois closure
| Galois group: | $F_5$ (as 10T4) |
| Inertia group: | $F_5$ |
| Unramified degree: | $1$ |
| Tame degree: | $4$ |
| Wild slopes: | [7/4] |
| Galois mean slope: | $31/20$ |
| Galois splitting model: | $x^{10} - x^{5} - 1$ |