Defining polynomial
|
\(x^{18} + x^{12} + x^{11} + x^{10} + x^{9} + 2 x^{8} + 2 x^{6} + x^{5} + 2 x^{3} + 2 x^{2} + 2\)
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $18$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $18$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 5 }) }$: | $18$ |
| This field is Galois and abelian over $\Q_{5}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{5}(\sqrt{2})$, 5.3.0.1, 5.6.0.1, 5.9.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 5.18.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{18} + x^{12} + x^{11} + x^{10} + x^{9} + 2 x^{8} + 2 x^{6} + x^{5} + 2 x^{3} + 2 x^{2} + 2 \)
|
| Relative Eisenstein polynomial: |
\( x - 5 \)
$\ \in\Q_{5}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
| Galois group: | $C_{18}$ (as 18T1) |
| Inertia group: | trivial |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $18$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: | Not computed |