Defining polynomial
|
\(x^{15} - 60 x^{12} + 15 x^{10} + 900 x^{9} - 300 x^{8} - 675 x^{7} + 9000 x^{6} + 22575 x^{5} + 14250 x^{4} - 125 x^{3} - 1875 x^{2} + 125\)
|
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: | $-1$ |
| $\card{ \Aut(K/\Q_{ 5 }) }$: | $1$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible slopes: | $[5/4]$ |
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
\( x^{3} + 3 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{5} + 10 t^{2} x^{2} + \left(5 t^{2} + 10\right) x + 5 \)
$\ \in\Q_{5}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 4t^{2} + 3$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |