# Properties

 Label 3.8.7.2 Base $$\Q_{3}$$ Degree $$8$$ e $$8$$ f $$1$$ c $$7$$ Galois group $QD_{16}$ (as 8T8)

# Related objects

## Defining polynomial

 $$x^{8} + 6$$ x^8 + 6

## Invariants

 Base field: $\Q_{3}$ Degree $d$: $8$ Ramification exponent $e$: $8$ Residue field degree $f$: $1$ Discriminant exponent $c$: $7$ Discriminant root field: $\Q_{3}(\sqrt{3\cdot 2})$ Root number: $-i$ $\card{ \Aut(K/\Q_{ 3 }) }$: $2$ This field is not Galois over $\Q_{3}.$ Visible slopes: None

## 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: $\Q_{3}$ Relative Eisenstein polynomial: $$x^{8} + 6$$ x^8 + 6

## Ramification polygon

 Residual polynomials: $z^{7} + 2z^{6} + z^{5} + 2z^{4} + z^{3} + 2z^{2} + z + 2$ Associated inertia: $2$ Indices of inseparability: $[0]$

## Invariants of the Galois closure

 Galois group: $\SD_{16}$ (as 8T8) Inertia group: $C_8$ (as 8T1) Wild inertia group: $C_1$ Unramified degree: $2$ Tame degree: $8$ Wild slopes: None Galois mean slope: $7/8$ Galois splitting model: $x^{8} - 2 x^{7} + x^{6} - 5 x^{5} + 7 x^{4} - 2 x^{3} + x^{2} - 5 x + 1$