Properties

Label 3.9.13.6
Base \(\Q_{3}\)
Degree \(9\)
e \(9\)
f \(1\)
c \(13\)
Galois group $(C_3^2:C_8):C_2$ (as 9T19)

Related objects

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Defining polynomial

\( x^{9} + 3 x^{5} + 3 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $9$
Ramification exponent $e$ : $9$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $13$
Discriminant root field: $\Q_{3}(\sqrt{3*})$
Root number: $-i$
$|\Aut(K/\Q_{ 3 })|$: $1$
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:\( x^{9} + 3 x^{5} + 3 \)

Invariants of the Galois closure

Galois group:$PSU(3,2):C_2$ (as 9T19)
Inertia group:$C_3^2:C_8$
Unramified degree:$2$
Tame degree:$8$
Wild slopes:[13/8, 13/8]
Galois mean slope:$37/24$
Galois splitting model:$x^{9} - 6 x^{7} - 12 x^{6} + 12 x^{5} + 60 x^{4} + 6 x^{3} - 48 x^{2} + 9 x + 8$