Defining polynomial
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$( x^{4} + x + 1 )^{4} + \left(2 x^{2} + 2 x + 2\right) ( x^{4} + x + 1 )^{3} + \left(2 x^{2} + 2 x\right) ( x^{4} + x + 1 )^{2} + 2$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $24$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $D_4$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2, 2]$ |
| Visible Swan slopes: | $[1,1]$ |
| Means: | $\langle\frac{1}{2}, \frac{3}{4}\rangle$ |
| Rams: | $(1, 1)$ |
| Jump set: | $[1, 3, 6]$ |
| Roots of unity: | $60 = (2^{ 4 } - 1) \cdot 2^{ 2 }$ |
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, 2.4.2.8a1.1, 2.4.2.8a5.2 x2, 2.2.4.12a6.1 x2 |
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
\( x^{4} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{4} + \left(2 t^{2} + 2 t\right) x^{3} + \left(2 t^{2} + 2 t\right) x^{2} + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^3 + (t^2 + t) z + (t^2 + t)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[3, 2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $32$ |
| Galois group: | $C_2^3:C_4$ (as 16T53) |
| Inertia group: | Intransitive group isomorphic to $C_2^3$ |
| Wild inertia group: | $C_2^3$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, 2]$ |
| Galois Swan slopes: | $[1,1,1]$ |
| Galois mean slope: | $1.75$ |
| Galois splitting model: | $x^{16} - 2 x^{15} + 2 x^{14} - 2 x^{13} + 28 x^{12} - 62 x^{11} + 70 x^{10} - 30 x^{9} + 6 x^{8} - 30 x^{7} + 70 x^{6} - 62 x^{5} + 28 x^{4} - 2 x^{3} + 2 x^{2} - 2 x + 1$ |