Properties

Label 17.9.8.1
Base \(\Q_{17}\)
Degree \(9\)
e \(9\)
f \(1\)
c \(8\)
Galois group $D_{9}$ (as 9T3)

Related objects

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Defining polynomial

\( x^{9} - 17 \)

Invariants

Base field: $\Q_{17}$
Degree $d$ : $9$
Ramification exponent $e$ : $9$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $8$
Discriminant root field: $\Q_{17}$
Root number: $1$
$|\Aut(K/\Q_{ 17 })|$: $1$
This field is not Galois over $\Q_{17}$.

Intermediate fields

17.3.2.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{17}$
Relative Eisenstein polynomial:\( x^{9} - 17 \)

Invariants of the Galois closure

Galois group:$D_9$ (as 9T3)
Inertia group:$C_9$
Unramified degree:$2$
Tame degree:$9$
Wild slopes:None
Galois mean slope:$8/9$
Galois splitting model:Not computed