Properties

Label 3.8.6.3
Base \(\Q_{3}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(6\)
Galois group $C_8:C_2$ (as 8T7)

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Defining polynomial

\(x^{8} - 3 x^{4} + 18\)  Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $8$
Ramification exponent $e$: $4$
Residue field degree $f$: $2$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{3}(\sqrt{2})$
Root number: $-1$
$|\Aut(K/\Q_{ 3 })|$: $4$
This field is not Galois over $\Q_{3}.$

Intermediate fields

$\Q_{3}(\sqrt{2})$, 3.4.2.2

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

Unramified/totally ramified tower

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

Invariants of the Galois closure

Galois group:$OD_{16}$ (as 8T7)
Inertia group:Intransitive group isomorphic to $C_4$
Wild inertia group:$C_1$
Unramified degree:$4$
Tame degree:$4$
Wild slopes:None
Galois mean slope:$3/4$
Galois splitting model:$x^{8} - x^{7} + 2 x^{6} + 2 x^{5} - 5 x^{4} + 13 x^{3} - 13 x^{2} + x + 1$