Properties

Label 3.8.4.1
Base \(\Q_{3}\)
Degree \(8\)
e \(2\)
f \(4\)
c \(4\)
Galois group $C_4\times C_2$ (as 8T2)

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Defining polynomial

\( x^{8} + 36 x^{4} - 27 x^{2} + 324 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $8$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $4$
Discriminant exponent $c$ : $4$
Discriminant root field: $\Q_{3}$
Root number: $-1$
$|\Gal(K/\Q_{ 3 })|$: $8$
This field is Galois and abelian over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{*})$, $\Q_{3}(\sqrt{3})$, $\Q_{3}(\sqrt{3*})$, 3.4.0.1, 3.4.2.1, 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:3.4.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{4} - x + 2 \)
Relative Eisenstein polynomial:$ x^{2} - 3 t^{2} \in\Q_{3}(t)[x]$

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:\( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \)