Properties

Label 3.9.10.2
Base \(\Q_{3}\)
Degree \(9\)
e \(9\)
f \(1\)
c \(10\)
Galois group $S_3^2:C_2$ (as 9T16)

Related objects

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Defining polynomial

\( x^{9} + 3 x^{2} + 6 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $9$
Ramification exponent $e$ : $9$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $10$
Discriminant root field: $\Q_{3}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 3 })|$: $1$
This field is not Galois over $\Q_{3}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 3 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}$
Relative Eisenstein polynomial:\( x^{9} + 3 x^{2} + 6 \)

Invariants of the Galois closure

Galois group:$S_3\wr C_2$ (as 9T16)
Inertia group:$C_3^2:C_4$
Unramified degree:$2$
Tame degree:$4$
Wild slopes:[5/4, 5/4]
Galois mean slope:$43/36$
Galois splitting model:$x^{9} - 3 x^{8} + 6 x^{7} - 6 x^{6} + 12 x^{4} - 18 x^{3} + 18 x^{2} - 9 x + 7$