Properties

Label 2.8.20.1
Base \(\Q_{2}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(20\)
Galois group $D_4\times C_2$ (as 8T9)

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Defining polynomial

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

Invariants

Base field: $\Q_{2}$
Degree $d$ : $8$
Ramification exponent $e$ : $4$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $20$
Discriminant root field: $\Q_{2}$
Root number: $-1$
$|\Aut(K/\Q_{ 2 })|$: $4$
This field is not Galois over $\Q_{2}$.

Intermediate fields

$\Q_{2}(\sqrt{*})$, $\Q_{2}(\sqrt{2})$, $\Q_{2}(\sqrt{2*})$, 2.4.6.1, 2.4.10.3, 2.4.10.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}(\sqrt{*})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} - x + 1 \)
Relative Eisenstein polynomial:$ x^{4} + \left(12 t + 12\right) x^{3} + 2 t + 14 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2\times D_4$ (as 8T9)
Inertia group:Intransitive group isomorphic to $D_4$
Unramified degree:$2$
Tame degree:$1$
Wild slopes:[2, 3, 7/2]
Galois mean slope:$11/4$
Galois splitting model:$x^{8} - 2 x^{6} + 5 x^{4} + 2 x^{2} + 1$