Defining polynomial
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$( x^{2} + 4 x + 2 )^{10} + 15 ( x^{2} + 4 x + 2 )^{4} + 5$
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Invariants
| Base field: | $\Q_{5}$ |
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| Degree $d$: | $20$ |
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| Ramification index $e$: | $10$ |
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| Residue field degree $f$: | $2$ |
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| Discriminant exponent $c$: | $26$ |
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| Discriminant root field: | $\Q_{5}$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{5})$ $=$ $\Gal(K/\Q_{5})$: | $D_{10}$ | |
| This field is Galois over $\Q_{5}.$ | ||
| Visible Artin slopes: | $[\frac{3}{2}]$ | |
| Visible Swan slopes: | $[\frac{1}{2}]$ | |
| Means: | $\langle\frac{2}{5}\rangle$ | |
| Rams: | $(1)$ | |
| Jump set: | undefined | |
| Roots of unity: | $24 = (5^{ 2 } - 1)$ |
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Intermediate fields
| $\Q_{5}(\sqrt{2})$, $\Q_{5}(\sqrt{5})$, $\Q_{5}(\sqrt{5\cdot 2})$, 5.2.2.2a1.2, 5.1.5.6a1.2 x5, 5.2.5.12a4.1 x5, 5.1.10.13a3.1, 5.1.10.13a2.2 x5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{2} + 4 x + 2 \)
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| Relative Eisenstein polynomial: |
\( x^{10} + 15 x^{4} + 5 \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^5 + 2$,$2 z^4 + 3$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $20$ |
| Galois group: | $D_{10}$ (as 20T4) |
| Inertia group: | Intransitive group isomorphic to $D_5$ |
| Wild inertia group: | $C_5$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $2$ |
| Galois Artin slopes: | $[\frac{3}{2}]$ |
| Galois Swan slopes: | $[\frac{1}{2}]$ |
| Galois mean slope: | $1.3$ |
| Galois splitting model: | $x^{20} + 5 x^{18} - 10 x^{16} - 48 x^{15} - 190 x^{14} - 440 x^{13} + 1005 x^{12} + 2360 x^{11} - 655 x^{10} - 2040 x^{9} - 340 x^{8} - 8840 x^{7} + 7480 x^{6} + 2704 x^{5} + 2800 x^{4} + 480 x^{3} + 80 x^{2} - 320 x + 64$ |