# Properties

 Label 3.9.9.6 Base $$\Q_{3}$$ Degree $$9$$ e $$3$$ f $$3$$ c $$9$$ Galois group $S_3\times C_3$ (as 9T4)

# Related objects

## Defining polynomial

 $$x^{9} + 3 x^{7} + 3 x^{6} + 18 x^{4} + 54$$

## Invariants

 Base field: $\Q_{3}$ Degree $d$ : $9$ Ramification exponent $e$ : $3$ Residue field degree $f$ : $3$ Discriminant exponent $c$ : $9$ Discriminant root field: $\Q_{3}(\sqrt{3*})$ Root number: $-i$ $|\Aut(K/\Q_{ 3 })|$: $3$ This field is not Galois 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: $x^{3} + \left(6 t + 6\right) x^{2} + \left(6 t^{2} + 6\right) x + 6 t^{2} + 3 \in\Q_{3}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_3\times S_3$ (as 9T4) Inertia group: Intransitive group isomorphic to $S_3$ Unramified degree: $3$ Tame degree: $2$ Wild slopes: [3/2] Galois mean slope: $7/6$ Galois splitting model: $x^{9} - 4 x^{6} + 3 x^{3} + 1$