Base \(\Q_{2}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(6\)
Galois group $C_2^2$ (as 4T2)

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Defining polynomial

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


Base field: $\Q_{2}$
Degree $d$: $4$
Ramification exponent $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{2}$
Root number: $-1$
$|\Gal(K/\Q_{ 2 })|$: $4$
This field is Galois and abelian over $\Q_{2}.$

Intermediate fields

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

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} - x + 1 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{2} + 6 t - 6 \)$\ \in\Q_{2}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2^2$ (as 4T2)
Inertia group:Intransitive group isomorphic to $C_2$
Wild inertia group:$C_2$
Unramified degree:$2$
Tame degree:$1$
Wild slopes:[3]
Galois mean slope:$3/2$
Galois splitting model:$x^{4} - 6 x^{2} + 4$