Properties

Label 2.4.9.1
Base \(\Q_{2}\)
Degree \(4\)
e \(4\)
f \(1\)
c \(9\)
Galois group $D_{4}$ (as 4T3)

Related objects

Learn more about

Defining polynomial

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

Invariants

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: $1$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}$.

Intermediate fields

$\Q_{2}(\sqrt{-1})$

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:\( x^{4} + 6 x^{2} + 2 \)

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