# Properties

 Label 3.9.9.4 Base $$\Q_{3}$$ Degree $$9$$ e $$3$$ f $$3$$ c $$9$$ Galois group $(C_3^2:C_3):C_2$ (as 9T22)

# Related objects

## Defining polynomial

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

## 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: $i$ $|\Aut(K/\Q_{ 3 })|$: $1$ 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: $x^{3} + 6 t x^{2} + 6 x + 6 t^{2} + 3 \in\Q_{3}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_3^3:C_6$ (as 9T22) Inertia group: Intransitive group isomorphic to $C_3^3:C_2$ Unramified degree: $3$ Tame degree: $2$ Wild slopes: [3/2, 3/2, 3/2] Galois mean slope: $79/54$ Galois splitting model: $x^{9} + 3 x^{7} - 18 x^{5} - 21 x^{4} - 34 x^{3} - 63 x^{2} - 42 x - 7$