Properties

Label 3.9.12.21
Base \(\Q_{3}\)
Degree \(9\)
e \(9\)
f \(1\)
c \(12\)
Galois group $C_3^2:C_4$ (as 9T9)

Related objects

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Defining polynomial

\( x^{9} + 3 x^{4} + 6 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $9$
Ramification exponent $e$ : $9$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{3}$
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:\( x^{9} + 3 x^{4} + 6 \)

Invariants of the Galois closure

Galois group:$C_3:S_3.C_2$ (as 9T9)
Inertia group:$C_3^2:C_2$
Unramified degree:$2$
Tame degree:$2$
Wild slopes:[3/2, 3/2]
Galois mean slope:$25/18$
Galois splitting model:$x^{9} - 3 x^{8} + 6 x^{7} - 18 x^{6} + 45 x^{5} - 63 x^{4} + 72 x^{3} - 72 x^{2} + 54 x - 18$