Defining polynomial
\(x^{12} + 40\)  |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $11$ |
| Discriminant root field: | $\Q_{5}(\sqrt{5\cdot 2})$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 5 })|$: | $4$ |
| This field is not Galois over $\Q_{5}.$ | |
Intermediate fields
| $\Q_{5}(\sqrt{5\cdot 2})$, 5.3.2.1, 5.4.3.4, 5.6.5.2 |
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: | \( x^{12} + 40 \) |