Properties

Label 2.8.31.17
Base \(\Q_{2}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(31\)
Galois group $C_8:C_2$ (as 8T7)

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Defining polynomial

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

Invariants

Base field: $\Q_{2}$
Degree $d$ : $8$
Ramification exponent $e$ : $8$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $31$
Discriminant root field: $\Q_{2}(\sqrt{2})$
Root number: $-1$
$|\Aut(K/\Q_{ 2 })|$: $4$
This field is not Galois over $\Q_{2}$.

Intermediate fields

$\Q_{2}(\sqrt{2})$, 2.4.11.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}$
Relative Eisenstein polynomial:\( x^{8} + 16 x^{5} + 12 x^{4} + 16 x^{3} + 2 \)

Invariants of the Galois closure

Galois group:$OD_{16}$ (as 8T7)
Inertia group:$C_8:C_2$
Unramified degree:$1$
Tame degree:$1$
Wild slopes:[2, 3, 4, 5]
Galois mean slope:$4$
Galois splitting model:\( x^{8} - 24 x^{6} - 108 x^{4} + 162 \)