Base \(\Q_{17}\)
Degree \(6\)
e \(6\)
f \(1\)
c \(5\)
Galois group $D_{6}$ (as 6T3)

Related objects

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Defining polynomial

\( x^{6} - 17 \)


Base field: $\Q_{17}$
Degree $d$: $6$
Ramification exponent $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $5$
Discriminant root field: $\Q_{17}(\sqrt{17})$
Root number: $-1$
$|\Aut(K/\Q_{ 17 })|$: $2$
This field is not Galois over $\Q_{17}.$

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:$\Q_{17}$
Relative Eisenstein polynomial:\( x^{6} - 17 \)

Invariants of the Galois closure

Galois group:$D_6$ (as 6T3)
Inertia group:$C_6$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:$x^{6} - 17$