Defining polynomial
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$( x^{4} + x + 1 )^{2} + 2 ( x^{4} + x + 1 ) + 2$
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Invariants
| Base field: | $\Q_{2}$ |
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| Degree $d$: | $8$ |
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| Ramification index $e$: | $2$ |
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| Residue field degree $f$: | $4$ |
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| Discriminant exponent $c$: | $8$ |
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| Discriminant root field: | $\Q_{2}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{2})$ $=$ $\Gal(K/\Q_{2})$: | $C_2\times C_4$ | |
| This field is Galois and abelian over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[2]$ | |
| Visible Swan slopes: | $[1]$ | |
| Means: | $\langle\frac{1}{2}\rangle$ | |
| Rams: | $(1)$ | |
| Jump set: | $[1, 2]$ | |
| Roots of unity: | $60 = (2^{ 4 } - 1) \cdot 2^{ 2 }$ |
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Intermediate fields
| $\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{-5})$, 2.4.1.0a1.1, 2.2.2.4a1.1, 2.2.2.4a1.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 2.4.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{4} + x + 1 \)
|
|
| Relative Eisenstein polynomial: |
\( x^{2} + 2 x + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois degree: | $8$ |
| Galois group: | $C_2\times C_4$ (as 8T2) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_2$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2]$ |
| Galois Swan slopes: | $[1]$ |
| Galois mean slope: | $1.0$ |
| Galois splitting model: | $x^{8} - x^{6} + x^{4} - x^{2} + 1$ |