Properties

Label 2.8.16.5
Base \(\Q_{2}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(16\)
Galois group $D_4$ (as 8T4)

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Defining polynomial

\(x^{8} + 4 x^{7} + 14 x^{6} + 36 x^{5} + 73 x^{4} + 88 x^{3} + 48 x^{2} + 56 x + 61\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $8$
Ramification exponent $e$: $4$
Residue field degree $f$: $2$
Discriminant exponent $c$: $16$
Discriminant root field: $\Q_{2}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 2 }) }$: $8$
This field is Galois over $\Q_{2}.$
Visible slopes:$[2, 3]$

Intermediate fields

$\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{-5})$, 2.4.4.1, 2.4.6.6 x2, 2.4.8.5 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: \( x^{4} + 2 x^{2} + 4 x + 6 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 1$,$z^{2} + 1$
Associated inertia:$1$,$1$
Indices of inseparability:$[5, 2, 0]$

Invariants of the Galois closure

Galois group:$D_4$ (as 8T4)
Inertia group:Intransitive group isomorphic to $C_2^2$
Wild inertia group:$C_2^2$
Unramified degree:$2$
Tame degree:$1$
Wild slopes:$[2, 3]$
Galois mean slope:$2$
Galois splitting model: $x^{8} + 4 x^{6} + 40 x^{2} + 4$ Copy content Toggle raw display