Properties

Label 2.8.22.1
Base \(\Q_{2}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(22\)
Galois group $D_4$ (as 8T4)

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Defining polynomial

\(x^{8} + 16 x^{7} + 68 x^{6} + 24 x^{5} + 48 x^{4} + 128 x^{3} + 280 x^{2} + 144 x + 108\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $8$
Ramification exponent $e$: $4$
Residue field degree $f$: $2$
Discriminant exponent $c$: $22$
Discriminant root field: $\Q_{2}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 2 }) }$: $8$
This field is Galois over $\Q_{2}.$
Visible slopes:$[3, 4]$

Intermediate fields

$\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{-2\cdot 5})$, 2.4.6.2, 2.4.11.13 x2, 2.4.11.18 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} + x + 1 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{4} + 8 x^{3} + \left(12 t + 8\right) x^{2} + 8 t x + 12 t + 6 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 1$,$z^{2} + 1$
Associated inertia:$1$,$1$
Indices of inseparability:$[8, 4, 0]$

Invariants of the Galois closure

Galois group:$D_4$ (as 8T4)
Inertia group:Intransitive group isomorphic to $C_4$
Wild inertia group:$C_4$
Unramified degree:$2$
Tame degree:$1$
Wild slopes:$[3, 4]$
Galois mean slope:$11/4$
Galois splitting model:$x^{8} + 4 x^{6} + 10 x^{4} + 24 x^{2} + 36$