Defining polynomial
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$( x^{5} + 4 x + 3 )^{3} + 5$
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Invariants
| Base field: | $\Q_{5}$ |
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| Degree $d$: | $15$ |
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| Ramification index $e$: | $3$ |
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| Residue field degree $f$: | $5$ |
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| Discriminant exponent $c$: | $10$ |
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| Discriminant root field: | $\Q_{5}(\sqrt{2})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{5})$: | $C_5$ | |
| This field is not Galois over $\Q_{5}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $3124 = (5^{ 5 } - 1)$ |
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Intermediate fields
| 5.1.3.2a1.1, 5.5.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 5.5.1.0a1.1 $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{5} + 4 x + 3 \)
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| Relative Eisenstein polynomial: |
\( x^{3} + 5 \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^2 + 3 z + 3$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $30$ |
| Galois group: | $C_5\times S_3$ (as 15T4) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $10$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.6666666666666666$ |
| Galois splitting model: |
$x^{15} - 2 x^{14} - 3 x^{13} - 6 x^{12} + 46 x^{11} - 96 x^{10} + 66 x^{9} - 271 x^{8} + 802 x^{7} - 1192 x^{6} + 2483 x^{5} - 2152 x^{4} + 5024 x^{3} - 605 x^{2} - 3875 x - 131$
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