Defining polynomial
|
\(x^{15} + 2 x^{5} + 3 x^{3} + 3 x^{2} + 4 x + 3\)
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $15$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{5}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 5 }) }$: | $15$ |
| This field is Galois and abelian over $\Q_{5}.$ | |
| Visible slopes: | None |
Intermediate fields
| 5.3.0.1, 5.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 5.15.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{15} + 2 x^{5} + 3 x^{3} + 3 x^{2} + 4 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x - 5 \)
$\ \in\Q_{5}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.