Properties

Label 2.4.4.3
Base \(\Q_{2}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(4\)
Galois group $D_{4}$ (as 4T3)

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Defining polynomial

\(x^{4} + 2 x^{3} + 4 x^{2} + 12 x + 12\) Copy content Toggle raw display

Invariants

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

Intermediate fields

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

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 \( x^{2} + x + 1 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + \left(2 t + 2\right) x + 4 t + 2 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Data not computed

Invariants of the Galois closure

Galois group: $D_4$ (as 4T3)
Inertia group: Intransitive group isomorphic to $C_2^2$
Wild inertia group: $C_2^2$
Unramified degree: $2$
Tame degree: $1$
Wild slopes: $[2, 2]$
Galois mean slope: $3/2$
Galois splitting model:$x^{4} - x^{2} - 1$