Properties

Label 3.9.12.25
Base \(\Q_{3}\)
Degree \(9\)
e \(9\)
f \(1\)
c \(12\)
Galois group $C_3^2:C_8$ (as 9T15)

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Defining polynomial

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

Invariants

Base field: $\Q_{3}$
Degree $d$: $9$
Ramification exponent $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
Discriminant root field: $\Q_{3}(\sqrt{2})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 3 }) }$: $1$
This field is not Galois over $\Q_{3}.$
Visible slopes:$[3/2, 3/2]$

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

Ramification polygon

Residual polynomials:$z^{4} + z + 2$
Associated inertia:$4$
Indices of inseparability:$[4, 3, 0]$

Invariants of the Galois closure

Galois group:$F_9$ (as 9T15)
Inertia group:$C_3:S_3$ (as 9T5)
Wild inertia group:$C_3^2$
Unramified degree:$4$
Tame degree:$2$
Wild slopes:$[3/2, 3/2]$
Galois mean slope:$25/18$
Galois splitting model:$x^{9} + 12 x^{7} - 24 x^{6} + 18 x^{5} + 24 x^{4} - 60 x^{3} - 72 x^{2} - 33 x - 8$