# Properties

 Label 5.9.8.1 Base $$\Q_{5}$$ Degree $$9$$ e $$9$$ f $$1$$ c $$8$$ Galois group $(C_9:C_3):C_2$ (as 9T10)

# Related objects

## Defining polynomial

 $$x^{9} - 5$$

## Invariants

 Base field: $\Q_{5}$ Degree $d$ : $9$ Ramification exponent $e$ : $9$ Residue field degree $f$ : $1$ Discriminant exponent $c$ : $8$ Discriminant root field: $\Q_{5}$ Root number: $1$ $|\Aut(K/\Q_{ 5 })|$: $1$ This field is not Galois over $\Q_{5}$.

## 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: $\Q_{5}$ Relative Eisenstein polynomial: $$x^{9} - 5$$

## Invariants of the Galois closure

 Galois group: $D_9:C_3$ (as 9T10) Inertia group: $C_9$ Unramified degree: $6$ Tame degree: $9$ Wild slopes: None Galois mean slope: $8/9$ Galois splitting model: $x^{9} - 5$