Defining polynomial
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\(x^{20} + 2 x^{13} + x^{11} + x^{10} + x^{9} + x^{8} + 2 x^{5} + 2 x^{4} + 2 x^{3} + x + 2\)
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Invariants
| Base field: | $\Q_{3}$ |
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| Degree $d$: | $20$ |
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| Ramification index $e$: | $1$ |
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| Residue field degree $f$: | $20$ |
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| Discriminant exponent $c$: | $0$ |
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| Discriminant root field: | $\Q_{3}(\sqrt{2})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{3})$ $=$ $\Gal(K/\Q_{3})$: | $C_{20}$ | |
| This field is Galois and abelian over $\Q_{3}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $3486784400 = (3^{ 20 } - 1)$ |
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Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.4.1.0a1.1, 3.5.1.0a1.1, 3.10.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 3.20.1.0a1.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{20} + 2 x^{13} + x^{11} + x^{10} + x^{9} + x^{8} + 2 x^{5} + 2 x^{4} + 2 x^{3} + x + 2 \)
|
|
| Relative Eisenstein polynomial: |
\( x - 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
| Galois degree: | $20$ |
| Galois group: | $C_{20}$ (as 20T1) |
| Inertia group: | trivial |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $20$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.0$ |
| Galois splitting model: | not computed |