Defining polynomial
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\(x^{15} + 5 x^{6} + 6 x^{5} + 6 x^{4} + 4 x^{3} + x^{2} + 2 x + 4\)
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Invariants
| Base field: | $\Q_{7}$ |
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| Degree $d$: | $15$ |
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| Ramification index $e$: | $1$ |
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| Residue field degree $f$: | $15$ |
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| Discriminant exponent $c$: | $0$ |
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| Discriminant root field: | $\Q_{7}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{7})$ $=$ $\Gal(K/\Q_{7})$: | $C_{15}$ | |
| This field is Galois and abelian over $\Q_{7}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $4747561509942 = (7^{ 15 } - 1)$ |
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Intermediate fields
| 7.3.1.0a1.1, 7.5.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 7.15.1.0a1.1 $\cong \Q_{7}(t)$ where $t$ is a root of
\( x^{15} + 5 x^{6} + 6 x^{5} + 6 x^{4} + 4 x^{3} + x^{2} + 2 x + 4 \)
|
|
| Relative Eisenstein polynomial: |
\( x - 7 \)
$\ \in\Q_{7}(t)[x]$
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Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
| Galois degree: | $15$ |
| Galois group: | $C_{15}$ (as 15T1) |
| Inertia group: | trivial |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $15$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.0$ |
| Galois splitting model: | not computed |