Properties

Label 3.9.22.10
Base \(\Q_{3}\)
Degree \(9\)
e \(9\)
f \(1\)
c \(22\)
Galois group $D_{9}$ (as 9T3)

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Defining polynomial

\(x^{9} + 21 x^{6} + 18 x^{5} + 75\) Copy content Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $9$
Ramification exponent $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $22$
Discriminant root field: $\Q_{3}$
Root number: $1$
$\card{ \Aut(K/\Q_{ 3 }) }$: $1$
This field is not Galois over $\Q_{3}.$
Visible slopes:$[2, 3]$

Intermediate fields

3.3.4.4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}$
Relative Eisenstein polynomial: \( x^{9} + 21 x^{6} + 18 x^{5} + 75 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{2} + 1$,$z^{6} + 1$
Associated inertia:$2$,$2$
Indices of inseparability:$[14, 6, 0]$

Invariants of the Galois closure

Galois group:$D_9$ (as 9T3)
Inertia group:$C_9$ (as 9T1)
Wild inertia group:$C_9$
Unramified degree:$2$
Tame degree:$1$
Wild slopes:$[2, 3]$
Galois mean slope:$22/9$
Galois splitting model:$x^{9} - 9 x^{7} + 27 x^{5} - 39 x^{3} + 36 x - 112$