Defining polynomial
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$( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + \left(8 x + 8\right) ( x^{2} + x + 1 ) + 2$
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Invariants
| Base field: | $\Q_{2}$ |
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| Degree $d$: | $8$ |
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| Ramification index $e$: | $4$ |
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| Residue field degree $f$: | $2$ |
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| Discriminant exponent $c$: | $22$ |
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| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ | |
| Root number: | $-i$ | |
| $\Aut(K/\Q_{2})$: | $C_2$ | |
| This field is not Galois over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[3, 4]$ | |
| Visible Swan slopes: | $[2,3]$ | |
| Means: | $\langle1, 2\rangle$ | |
| Rams: | $(2, 4)$ | |
| Jump set: | $[1, 3, 7]$ | |
| Roots of unity: | $6 = (2^{ 2 } - 1) \cdot 2$ |
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Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.2.2.6a1.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{2} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{4} + 8 t x^{3} + \left(4 t + 4\right) x^{2} + 8 t x + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[8, 4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $16$ |
| Galois group: | $\SD_{16}$ (as 8T8) |
| Inertia group: | Intransitive group isomorphic to $D_4$ |
| Wild inertia group: | $D_4$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 3, 4]$ |
| Galois Swan slopes: | $[1,2,3]$ |
| Galois mean slope: | $3.0$ |
| Galois splitting model: | $x^{8} - 10 x^{4} - 100$ |