Defining polynomial
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$( x^{2} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $8$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $22$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$ $=$$\Gal(K/\Q_{2})$: | $C_8$ |
| This field is Galois and abelian 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: | $6 = (2^{ 2 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.2.2.6a1.6 |
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: |
\( x^{4} + 8 t x^{3} + 4 x^{2} + 8 t + 2 \)
$\ \in\Q_{2}(t)[x]$
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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: | $8$ |
| Galois group: | $C_8$ (as 8T1) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Wild inertia group: | $C_4$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[3, 4]$ |
| Galois Swan slopes: | $[2,3]$ |
| Galois mean slope: | $2.75$ |
| Galois splitting model: | Does not exist |