# Properties

 Label 3.9.9.12 Base $$\Q_{3}$$ Degree $$9$$ e $$9$$ f $$1$$ c $$9$$ Galois group $(C_3^2:C_8):C_2$ (as 9T19)

# Related objects

## Defining polynomial

 $$x^{9} + 6 x + 3$$ x^9 + 6*x + 3

## Invariants

 Base field: $\Q_{3}$ Degree $d$: $9$ Ramification exponent $e$: $9$ Residue field degree $f$: $1$ Discriminant exponent $c$: $9$ Discriminant root field: $\Q_{3}(\sqrt{3\cdot 2})$ Root number: $-i$ $\card{ \Aut(K/\Q_{ 3 }) }$: $1$ This field is not Galois over $\Q_{3}.$ Visible slopes: $[9/8, 9/8]$

## 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: $$x^{9} + 6 x + 3$$ x^9 + 6*x + 3

## Ramification polygon

 Residual polynomials: $z + 1$ Associated inertia: $1$ Indices of inseparability: $[1, 1, 0]$

## Invariants of the Galois closure

 Galois group: $F_9:C_2$ (as 9T19) Inertia group: $F_9$ (as 9T15) Wild inertia group: $C_3^2$ Unramified degree: $2$ Tame degree: $8$ Wild slopes: $[9/8, 9/8]$ Galois mean slope: $79/72$ Galois splitting model: $x^{9} - 3 x^{8} + 12 x^{5} + 12 x^{4} - 12 x + 4$