Base \(\Q_{2}\)
Degree \(6\)
e \(6\)
f \(1\)
c \(8\)
Galois group $D_{6}$ (as 6T3)

Related objects

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Defining polynomial

\( x^{6} + 2 x^{3} + 2 \)


Base field: $\Q_{2}$
Degree $d$: $6$
Ramification exponent $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
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


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:\( x^{6} + 2 x^{3} + 2 \)

Invariants of the Galois closure

Galois group:$D_6$ (as 6T3)
Inertia group:$C_6$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:[2]
Galois mean slope:$4/3$
Galois splitting model:$x^{6} + 2 x^{3} + 2$