Properties

Label 2.4.6.9
Base \(\Q_{2}\)
Degree \(4\)
e \(4\)
f \(1\)
c \(6\)
Galois group $D_{4}$ (as 4T3)

Related objects

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Defining polynomial

\(x^{4} + 2 x^{3} + 6\)  Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $4$
Ramification exponent $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{2}(\sqrt{5})$
Root number: $1$
$|\Aut(K/\Q_{ 2 })|$: $2$
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:$\Q_{2}$
Relative Eisenstein polynomial:\( x^{4} + 2 x^{3} + 6 \)  Toggle raw display

Invariants of the Galois closure

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