Properties

Label 2.8.8.6
Base \(\Q_{2}\)
Degree \(8\)
e \(2\)
f \(4\)
c \(8\)
Galois group $(C_8:C_2):C_2$ (as 8T16)

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Defining polynomial

\( x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $8$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $4$
Discriminant exponent $c$ : $8$
Discriminant root field: $\Q_{2}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}$.

Intermediate fields

$\Q_{2}(\sqrt{*})$, 2.4.0.1

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

Unramified/totally ramified tower

Unramified subfield:2.4.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{4} - x + 1 \)
Relative Eisenstein polynomial:$ x^{2} + \left(2 t^{3} + 2 t + 2\right) x + 2 t^{3} + 2 t + 2 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2^3.C_4$ (as 8T16)
Inertia group:Intransitive group isomorphic to $C_2^3$
Unramified degree:$4$
Tame degree:$1$
Wild slopes:[2, 2, 2]
Galois mean slope:$7/4$
Galois splitting model:\( x^{8} - 5 x^{6} + 10 x^{4} - 10 x^{2} + 5 \)