Defining polynomial
|
\(x^{12} - 50 x^{6} - 175\)
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 5 }) }$: | $12$ |
| This field is Galois over $\Q_{5}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{5}(\sqrt{2})$, 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: | $\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{2} + 4 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{6} + 20 t + 15 \)
$\ \in\Q_{5}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{5} + z^{4} + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |