Defining polynomial
| \( x^{15} + 10 x^{14} + 15 x^{12} + 20 x^{11} + 2 x^{10} + 5 x^{9} + 10 x^{8} + 15 x^{7} + 15 x^{6} + 12 x^{5} + 20 x^{4} + 10 x^{3} + 10 x^{2} + 20 x + 12 \) |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $5$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{5}(\sqrt{5})$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 5 })|$: | $1$ |
| This field is not Galois over $\Q_{5}.$ | |
Intermediate fields
| 5.3.0.1 |
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^{5} + 20 x^{4} + \left(10 t^{2} + 10\right) x^{3} + \left(10 t^{2} + 20 t\right) x^{2} + \left(10 t^{2} + 10 t + 5\right) x + 10 t^{2} + 15 \in\Q_{5}(t)[x]$ |