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

\( x^{8} + 8 x^{5} + 6 x^{4} + 16 x^{3} + 8 x^{2} + 12 \)

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: $1$
$|\Gal(K/\Q_{ 2 })|$: $8$
This field is Galois over $\Q_{2}$.

Intermediate fields

$\Q_{2}(\sqrt{*})$, $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{-2*})$, 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:$\Q_{2}(\sqrt{*})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} - x + 1 \)
Relative Eisenstein polynomial:$ x^{4} + 8 x^{3} + \left(8 t + 4\right) x^{2} + \left(8 t + 8\right) x + 2 t + 2 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

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