Defining polynomial
| \( x^{12} + 25 x^{6} + 200 \) |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $6$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $10$ |
| Discriminant root field: | $\Q_{5}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 5 })|$: | $12$ |
| This field is Galois over $\Q_{5}$. | |
Intermediate fields
| $\Q_{5}(\sqrt{*})$, 5.3.2.1 x3, 5.4.2.2, 5.6.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{5}(\sqrt{*})$ $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{2} - x + 2 \) |
| Relative Eisenstein polynomial: | $ x^{6} - 5 t^{3} \in\Q_{5}(t)[x]$ |