Defining polynomial
| \( x^{10} - 5 x^{8} + 5 \) |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$ : | $10$ |
| Ramification exponent $e$ : | $10$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $17$ |
| Discriminant root field: | $\Q_{5}(\sqrt{5})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 5 })|$: | $10$ |
| This field is Galois and abelian over $\Q_{5}$. | |
Intermediate fields
| $\Q_{5}(\sqrt{5})$, 5.5.8.2 |
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: | \( x^{10} - 30 x^{8} + 225 x^{6} + 5 \) |