Properties

Label 2.4.6.5
Base \(\Q_{2}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(6\)
Galois group $D_{4}$ (as 4T3)

Related objects

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Defining polynomial

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

Invariants

Base field: $\Q_{2}$
Degree $d$: $4$
Ramification exponent $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{2}(\sqrt{-1})$
Root number: $i$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}.$

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 \( x^{2} - x + 1 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{2} + 6 t - 2 \)$\ \in\Q_{2}(t)[x]$  Toggle raw display

Invariants of the Galois closure

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