Base \(\Q_{2}\)
Degree \(4\)
e \(4\)
f \(1\)
c \(8\)
Galois group $C_2^2$ (as 4T2)

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Defining polynomial

\(x^{4} + 6 x^{2} + 1\)  Toggle raw display


Base field: $\Q_{2}$
Degree $d$: $4$
Ramification exponent $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Discriminant root field: $\Q_{2}$
Root number: $-1$
$|\Gal(K/\Q_{ 2 })|$: $4$
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:\( x^{4} + 4 x^{3} + 14 x^{2} - 52 x + 34 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2^2$ (as 4T2)
Inertia group:$C_2^2$
Wild inertia group:$C_2^2$
Unramified degree:$1$
Tame degree:$1$
Wild slopes:[2, 3]
Galois mean slope:$2$
Galois splitting model:$x^{4} + 6 x^{2} + 1$  Toggle raw display