Defining polynomial
|
$( x^{2} + x + 1 )^{4} + 2 ( x^{2} + x + 1 )^{2} + 4 x ( x^{2} + x + 1 ) + 2$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $8$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $16$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$: | $C_4$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2, 3]$ |
| Visible Swan slopes: | $[1,2]$ |
| Means: | $\langle\frac{1}{2}, \frac{5}{4}\rangle$ |
| Rams: | $(1, 3)$ |
| Jump set: | $[1, 5, 9]$ |
| Roots of unity: | $12 = (2^{ 2 } - 1) \cdot 2^{ 2 }$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{-5})$, 2.2.2.4a1.1 |
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
\( x^{2} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + \left(4 t + 4\right) x^{3} + 2 x^{2} + 4 t x + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^2 + 1$,$z + t$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[5, 2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $16$ |
| Galois group: | $D_4:C_2$ (as 8T11) |
| Inertia group: | Intransitive group isomorphic to $C_2\times C_4$ |
| Wild inertia group: | $C_2\times C_4$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 3, 3]$ |
| Galois Swan slopes: | $[1,2,2]$ |
| Galois mean slope: | $2.5$ |
| Galois splitting model: | $x^{8} - 2 x^{6} + 5 x^{4} - 4 x^{2} + 1$ |