Base \(\Q_{2}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(24\)
Galois group $C_4\times C_2$ (as 8T2)

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Defining polynomial

\(x^{8} + 16\)  Toggle raw display


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

Intermediate fields

$\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{2})$,,,

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:\( x^{8} + 24 x^{5} + 50 x^{4} + 64 x^{3} + 428 x^{2} + 776 x + 578 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2\times C_4$ (as 8T2)
Inertia group:$C_4\times C_2$
Wild inertia group:$C_2\times C_4$
Unramified degree:$1$
Tame degree:$1$
Wild slopes:[2, 3, 4]
Galois mean slope:$3$
Galois splitting model:$x^{8} + 16$  Toggle raw display