# Properties

 Label 3.9.12.3 Base $$\Q_{3}$$ Degree $$9$$ e $$3$$ f $$3$$ c $$12$$ Galois group $C_9$ (as 9T1)

# Learn more about

## Defining polynomial

 $$x^{9} + 6 x^{8} + 3 x^{6} + 9 x^{3} + 135$$

## Invariants

 Base field: $\Q_{3}$ Degree $d$ : $9$ Ramification exponent $e$ : $3$ Residue field degree $f$ : $3$ Discriminant exponent $c$ : $12$ Discriminant root field: $\Q_{3}$ Root number: $1$ $|\Gal(K/\Q_{ 3 })|$: $9$ This field is Galois and abelian over $\Q_{3}$.

## Intermediate fields

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

## Unramified/totally ramified tower

 Unramified subfield: 3.3.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of $$x^{3} - x + 1$$ Relative Eisenstein polynomial: $x^{3} + \left(3 t^{2} + 9 t\right) x^{2} + \left(9 t^{2} + 18 t + 18\right) x + 6 t^{2} + 6 t + 15 \in\Q_{3}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_9$ (as 9T1) Inertia group: Intransitive group isomorphic to $C_3$ Unramified degree: $3$ Tame degree: $1$ Wild slopes: [2] Galois mean slope: $4/3$ Galois splitting model: $x^{9} - 57 x^{7} - 38 x^{6} + 855 x^{5} + 228 x^{4} - 4902 x^{3} + 1710 x^{2} + 9063 x - 7201$