Properties

Label 2.8.22.83
Base \(\Q_{2}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(22\)
Galois group $D_4$ (as 8T4)

Related objects

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Defining polynomial

\( x^{8} + 4 x^{7} + 2 x^{4} + 4 x^{2} + 14 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $8$
Ramification exponent $e$ : $8$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $22$
Discriminant root field: $\Q_{2}$
Root number: $1$
$|\Gal(K/\Q_{ 2 })|$: $8$
This field is Galois over $\Q_{2}$.

Intermediate fields

$\Q_{2}(\sqrt{-*})$, $\Q_{2}(\sqrt{-2*})$, $\Q_{2}(\sqrt{2})$, 2.4.8.3, 2.4.9.7 x2, 2.4.10.6 x2

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:\( x^{8} + 4 x^{7} + 2 x^{4} + 4 x^{2} + 14 \)

Invariants of the Galois closure

Galois group:$D_4$ (as 8T4)
Inertia group:$D_4$
Unramified degree:$1$
Tame degree:$1$
Wild slopes:[2, 3, 7/2]
Galois mean slope:$11/4$
Galois splitting model:\( x^{8} - 4 x^{7} + 8 x^{6} - 4 x^{5} + 4 x^{4} + 4 x^{3} + 32 x^{2} - 20 x + 25 \)