Properties

Label 3.9.12.16
Base \(\Q_{3}\)
Degree \(9\)
e \(3\)
f \(3\)
c \(12\)
Galois group $S_3\times C_3$ (as 9T4)

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Defining polynomial

\( x^{9} + 9 x^{5} + 18 x^{3} + 27 x^{2} + 27 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $9$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $3$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{3}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 3 })|$: $3$
This field is not Galois over $\Q_{3}$.

Intermediate fields

3.3.0.1, 3.3.4.4

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

Unramified/totally ramified tower

Unramified subfield:3.3.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{3} - x + 1 \)
Relative Eisenstein polynomial:$ x^{3} + 6 t x^{2} + 6 t + 3 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3\times S_3$ (as 9T4)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$6$
Tame degree:$1$
Wild slopes:[2]
Galois mean slope:$4/3$
Galois splitting model:$x^{9} + 6 x^{7} - 8 x^{6} - 9 x^{5} - 60 x^{4} - 15 x^{3} - 18 x^{2} + 108 x + 8$