Properties

Label 3.9.12.26
Base \(\Q_{3}\)
Degree \(9\)
e \(9\)
f \(1\)
c \(12\)
Galois group $C_3^2:C_8$ (as 9T15)

Related objects

Learn more about

Defining polynomial

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

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: $1$
$|\Aut(K/\Q_{ 3 })|$: $1$
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:\( x^{9} + 3 x^{4} + 3 x^{3} + 3 \)

Invariants of the Galois closure

Galois group:$F_9$ (as 9T15)
Inertia group:$C_3^2:C_2$
Unramified degree:$4$
Tame degree:$2$
Wild slopes:[3/2, 3/2]
Galois mean slope:$25/18$
Galois splitting model:$x^{9} - 9 x^{7} - 21 x^{6} + 72 x^{5} + 99 x^{4} - 99 x^{3} - 585 x^{2} + 549 x + 166$