Properties

Label 2.8.24.12
Base \(\Q_{2}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(24\)
Galois group $Q_8$ (as 8T5)

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Defining polynomial

\(x^{8} + 14 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 30\)  Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $8$
Ramification exponent $e$: $8$
Residue field degree $f$: $1$
Discriminant exponent $c$: $24$
Discriminant root field: $\Q_{2}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 2 }) }$: $8$
This field is Galois over $\Q_{2}.$

Intermediate fields

$\Q_{2}(\sqrt{-5})$, $\Q_{2}(\sqrt{-2\cdot 5})$, $\Q_{2}(\sqrt{2})$, 2.4.8.3

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:\( x^{8} + 14 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 30 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$Q_8$ (as 8T5)
Inertia group:$Q_8$
Wild inertia group:$Q_8$
Unramified degree:$1$
Tame degree:$1$
Wild slopes:[2, 3, 4]
Galois mean slope:$3$
Galois splitting model:$x^{8} + 12 x^{6} + 36 x^{4} + 36 x^{2} + 9$