Properties

Label 3.6.8.9
Base \(\Q_{3}\)
Degree \(6\)
e \(3\)
f \(2\)
c \(8\)
Galois group $S_3\times C_3$ (as 6T5)

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Defining polynomial

\(x^{6} + 36 x^{4} + 6 x^{3} + 9\) Copy content Toggle raw display

Invariants

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

Intermediate fields

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

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 \( x^{2} + 2 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{3} + \left(6 t + 6\right) x^{2} + 3 \) $\ \in\Q_{3}(t)[x]$ Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_3\times S_3$ (as 6T5)
Inertia group:Intransitive group isomorphic to $C_3^2$
Wild inertia group:$C_3^2$
Unramified degree:$2$
Tame degree:$1$
Wild slopes:$[2, 2]$
Galois mean slope:$16/9$
Galois splitting model:$x^{6} + 6 x^{4} - 3 x^{3} + 9 x^{2} - 9 x + 4$