Properties

Label 3.9.13.1
Base \(\Q_{3}\)
Degree \(9\)
e \(9\)
f \(1\)
c \(13\)
Galois group $(C_3^2:C_8):C_2$ (as 9T19)

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Defining polynomial

\(x^{9} + 6 x^{5} + 3\) Copy content Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $9$
Ramification exponent $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $13$
Discriminant root field: $\Q_{3}(\sqrt{3})$
Root number: $i$
$\card{ \Aut(K/\Q_{ 3 }) }$: $1$
This field is not Galois over $\Q_{3}.$
Visible slopes:$[13/8, 13/8]$

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: \( x^{9} + 6 x^{5} + 3 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 2$
Associated inertia:$1$
Indices of inseparability:$[5, 5, 0]$

Invariants of the Galois closure

Galois group:$F_9:C_2$ (as 9T19)
Inertia group:$F_9$ (as 9T15)
Wild inertia group:$C_3^2$
Unramified degree:$2$
Tame degree:$8$
Wild slopes:$[13/8, 13/8]$
Galois mean slope:$37/24$
Galois splitting model:$x^{9} - 3 x^{8} - 18 x^{5} + 54 x^{4} + 432 x^{3} - 1728 x^{2} + 2187 x - 945$