$p$-adic field 2.8.16.6

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

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

\(x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36\)

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:	$1$
$\card{ \Gal(K/\Q_{ 2 }) }$:	$8$
This field is Galois and abelian over $\Q_{2}.$
Visible slopes:	$[2, 3]$

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 \( x^{2} + x + 1 \)
Relative Eisenstein polynomial:	\( x^{4} + \left(4 t + 4\right) x^{3} + \left(4 t + 6\right) x^{2} + 4 x + 6 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

Residual polynomials:	$z + 1$,$z^{2} + 1$
Associated inertia:	$1$,$1$
Indices of inseparability:	$[5, 2, 0]$

Invariants of the Galois closure

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