Properties

Label 2.8.16.6
Base \(\Q_{2}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(16\)
Galois group $C_2^3$ (as 8T3)

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This field has $7$ quadratic subextensions (the most among all $p$-adic fields), which is used in the hardest case of the Kronecker-Weber theorem ($p=2$). It is also the splitting field over $\mathbb{Q}_2$ of $x^8-16$, the counterexample that corrected the statement of the Grunwald-Wang theorem.

Defining polynomial

\(x^{8} + 4 x^{6} + 8 x^{2} + 4\)  Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $8$
Ramification exponent $e$: $4$
Residue field degree $f$: $2$
Discriminant exponent $c$: $16$
Discriminant root field: $\Q_{2}$
Root number: $1$
$|\Gal(K/\Q_{ 2 })|$: $8$
This field is Galois and abelian over $\Q_{2}.$

Intermediate fields

$\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{-5})$, $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{-2\cdot 5})$, $\Q_{2}(\sqrt{2})$, $\Q_{2}(\sqrt{2\cdot 5})$, 2.4.4.1, 2.4.6.1, 2.4.6.2, 2.4.8.4, 2.4.8.3, 2.4.8.2, 2.4.8.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} - x + 1 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{4} + \left(8 t + 4\right) x^{3} + 10 x^{2} + \left(8 t + 12\right) x + 14 \)$\ \in\Q_{2}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2^3$ (as 8T3)
Inertia group:Intransitive group isomorphic to $C_2^2$
Unramified degree:$2$
Tame degree:$1$
Wild slopes:[2, 3]
Galois mean slope:$2$
Galois splitting model:$x^{8} - x^{4} + 1$