Base \(\Q_{2}\)
Degree \(4\)
e \(4\)
f \(1\)
c \(9\)
Galois group $D_{4}$ (as 4T3)

Related objects

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Defining polynomial

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


Base field: $\Q_{2}$
Degree $d$: $4$
Ramification exponent $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $9$
Discriminant root field: $\Q_{2}(\sqrt{-2})$
Root number: $-i$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}.$

Intermediate fields


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

Invariants of the Galois closure

Galois group:$D_4$ (as 4T3)
Inertia group:$D_{4}$
Unramified degree:$1$
Tame degree:$1$
Wild slopes:[2, 3, 7/2]
Galois mean slope:$11/4$
Galois splitting model:$x^{4} - 2 x^{2} - 2$