Defining polynomial
|
\(x^{20} + x^{12} + 5 x^{11} + 16 x^{10} + 14 x^{9} + 13 x^{8} + 3 x^{7} + 14 x^{6} + 9 x^{5} + x^{4} + 13 x^{3} + 2 x^{2} + 5 x + 3\)
|
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $20$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $20$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{17}(\sqrt{3})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 17 }) }$: | $20$ |
| This field is Galois and abelian over $\Q_{17}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{17}(\sqrt{3})$, 17.4.0.1, 17.5.0.1, 17.10.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 17.20.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of
\( x^{20} + x^{12} + 5 x^{11} + 16 x^{10} + 14 x^{9} + 13 x^{8} + 3 x^{7} + 14 x^{6} + 9 x^{5} + x^{4} + 13 x^{3} + 2 x^{2} + 5 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x - 17 \)
$\ \in\Q_{17}(t)[x]$
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Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
| Galois group: | $C_{20}$ (as 20T1) |
| Inertia group: | trivial |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $20$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: | Not computed |