Defining polynomial
|
\(x^{8} + x^{4} + 65 x^{3} + 23 x^{2} + 42 x + 2\)
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Invariants
| Base field: | $\Q_{83}$ |
|
| Degree $d$: | $8$ |
|
| Ramification index $e$: | $1$ |
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| Residue field degree $f$: | $8$ |
|
| Discriminant exponent $c$: | $0$ |
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| Discriminant root field: | $\Q_{83}(\sqrt{2})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{83})$ $=$ $\Gal(K/\Q_{83})$: | $C_8$ | |
| This field is Galois and abelian over $\Q_{83}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $2252292232139040 = (83^{ 8 } - 1)$ |
|
Intermediate fields
| $\Q_{83}(\sqrt{2})$, 83.4.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 83.8.1.0a1.1 $\cong \Q_{83}(t)$ where $t$ is a root of
\( x^{8} + x^{4} + 65 x^{3} + 23 x^{2} + 42 x + 2 \)
|
|
| Relative Eisenstein polynomial: |
\( x - 83 \)
$\ \in\Q_{83}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.