Defining polynomial
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$( x^{2} + 4 x + 2 )^{5} + \left(50 x + 100\right) ( x^{2} + 4 x + 2 ) + 5$
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Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $10$ |
| Ramification index $e$: | $5$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $18$ |
| Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{5})$: | $C_1$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible Artin slopes: | $[\frac{9}{4}]$ |
| Visible Swan slopes: | $[\frac{5}{4}]$ |
| Means: | $\langle1\rangle$ |
| Rams: | $(\frac{5}{4})$ |
| Jump set: | undefined |
| Roots of unity: | $24 = (5^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{5}(\sqrt{2})$ |
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: |
\( x^{5} + \left(75 t + 25\right) x + 5 \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + (3 t + 1)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[5, 0]$ |
Invariants of the Galois closure
| Galois degree: | $200$ |
| Galois group: | $D_5:F_5$ (as 10T17) |
| Inertia group: | Intransitive group isomorphic to $C_5:F_5$ |
| Wild inertia group: | $C_5^2$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\frac{5}{4}, \frac{9}{4}]$ |
| Galois Swan slopes: | $[\frac{1}{4},\frac{5}{4}]$ |
| Galois mean slope: | $2.03$ |
| Galois splitting model: |
$x^{10} - 10 x^{5} + 1275$
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