Defining polynomial
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$( x^{2} + 4 x + 2 )^{10} + 5 ( x^{2} + 4 x + 2 )^{2} + 5$
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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 $c$: | $22$ |
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| Discriminant root field: | $\Q_{5}$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{5})$: | $C_2^2$ | |
| This field is not Galois over $\Q_{5}.$ | ||
| Visible Artin slopes: | $[\frac{5}{4}]$ | |
| Visible Swan slopes: | $[\frac{1}{4}]$ | |
| Means: | $\langle\frac{1}{5}\rangle$ | |
| Rams: | $(\frac{1}{2})$ | |
| 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, 5.1.5.5a1.1, 5.2.5.10a4.1, 5.1.10.11a2.1, 5.1.10.11a1.3 |
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: |
\( x^{10} + 5 x^{2} + 5 \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^5 + 2$,$2 z^2 + 3$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $40$ |
| Galois group: | $C_2\times F_5$ (as 20T13) |
| Inertia group: | Intransitive group isomorphic to $F_5$ |
| Wild inertia group: | $C_5$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\frac{5}{4}]$ |
| Galois Swan slopes: | $[\frac{1}{4}]$ |
| Galois mean slope: | not computed |
| Galois splitting model: |
$x^{20} - 5 x^{19} + 10 x^{18} - 15 x^{17} + 30 x^{16} - 48 x^{15} + 75 x^{14} - 165 x^{13} + 195 x^{12} + 25 x^{11} - 197 x^{10} + 25 x^{9} + 195 x^{8} - 165 x^{7} + 75 x^{6} - 48 x^{5} + 30 x^{4} - 15 x^{3} + 10 x^{2} - 5 x + 1$
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