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

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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

5.3.2.1

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:$x^{9} - 5$