Properties

Label 17.8.7.4
Base \(\Q_{17}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(7\)
Galois group $C_8$

Related objects

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Defining polynomial

\( x^{8} - 12393 \)

Invariants

Base field: $\Q_{17}$
Degree $d$ : $8$
Ramification exponent $e$ : $8$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $7$
Discriminant root field: $\Q_{17}(\sqrt{17})$
Root number: $1$
$|\Gal(K/\Q_{ 17 })|$: $8$
This field is Galois and abelian over $\Q_{17}$.

Intermediate fields

$\Q_{17}(\sqrt{17})$, 17.4.3.2

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:\( x^{8} - 12393 \)

Invariants of the Galois closure

Galois group:$C_8$
Inertia group:$C_8$
Unramified degree:$1$
Tame degree:$8$
Wild slopes:None
Galois Mean Slope:$7/8$
Global Splitting Model:Not computed