Properties

Label 2.8.8.12
Base \(\Q_{2}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(8\)
Galois group $A_4\wr C_2$ (as 8T42)

Related objects

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Defining polynomial

\( x^{8} + 2 x^{5} + 2 x^{4} + 4 \)

Invariants

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

Intermediate fields

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

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} + 2 x^{3} + 2 x^{2} + \left(2 t + 2\right) x + 2 t + 2 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$A_4\wr C_2$ (as 8T42)
Inertia group:Intransitive group isomorphic to $C_2^4:C_3$
Unramified degree:$6$
Tame degree:$3$
Wild slopes:[4/3, 4/3, 4/3, 4/3]
Galois mean slope:$31/24$
Galois splitting model:$x^{8} - 4 x^{7} + 4 x^{6} - 2 x^{5} + 14 x^{4} - 20 x^{3} - 4 x^{2} + 10 x + 5$