Properties

Label 2.4.10.3
Base \(\Q_{2}\)
Degree \(4\)
e \(4\)
f \(1\)
c \(10\)
Galois group $D_{4}$ (as 4T3)

Related objects

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Defining polynomial

\( x^{4} + 6 x^{2} - 9 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $4$
Ramification exponent $e$ : $4$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $10$
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

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

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:\( x^{4} - 4 x^{3} + 12 x^{2} - 16 x - 2 \)

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:$x^{4} + 6 x^{2} - 9$