Properties

Label 23.8.4.1
Base \(\Q_{23}\)
Degree \(8\)
e \(2\)
f \(4\)
c \(4\)
Galois group $C_4\times C_2$ (as 8T2)

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Defining polynomial

\(x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761\)  Toggle raw display

Invariants

Base field: $\Q_{23}$
Degree $d$: $8$
Ramification exponent $e$: $2$
Residue field degree $f$: $4$
Discriminant exponent $c$: $4$
Discriminant root field: $\Q_{23}$
Root number: $-1$
$|\Gal(K/\Q_{ 23 })|$: $8$
This field is Galois and abelian over $\Q_{23}.$

Intermediate fields

$\Q_{23}(\sqrt{5})$, $\Q_{23}(\sqrt{23})$, $\Q_{23}(\sqrt{23\cdot 5})$, 23.4.0.1, 23.4.2.1, 23.4.2.2

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

Unramified/totally ramified tower

Unramified subfield:23.4.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of \( x^{4} - x + 11 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{2} - 23 t^{2} \)$\ \in\Q_{23}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2\times C_4$ (as 8T2)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$4$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:Not computed