Defining polynomial
\(x^{12} - 25 x^{4} + 250\)  |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{5}(\sqrt{5\cdot 2})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 5 })|$: | $12$ |
| This field is Galois and abelian over $\Q_{5}.$ | |
Intermediate fields
| $\Q_{5}(\sqrt{5\cdot 2})$, 5.3.0.1, 5.4.3.4, 5.6.3.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 5.3.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{3} - x + 2 \) |
| Relative Eisenstein polynomial: | \( x^{4} - 5 t \)$\ \in\Q_{5}(t)[x]$ |