Properties

Label 17.15.14.1
Base \(\Q_{17}\)
Degree \(15\)
e \(15\)
f \(1\)
c \(14\)
Galois group $C_{15} : C_4$ (as 15T6)

Related objects

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Defining polynomial

\( x^{15} - 17 \)

Invariants

Base field: $\Q_{17}$
Degree $d$: $15$
Ramification exponent $e$: $15$
Residue field degree $f$: $1$
Discriminant exponent $c$: $14$
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, 17.5.4.1

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:\( x^{15} - 17 \)

Invariants of the Galois closure

Galois group:$C_3:F_5$ (as 15T6)
Inertia group:$C_{15}$
Unramified degree:$4$
Tame degree:$15$
Wild slopes:None
Galois mean slope:$14/15$
Galois splitting model:Not computed