Properties

Label 17.8.7.6
Base \(\Q_{17}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(7\)
Galois group $C_8$ (as 8T1)

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Defining polynomial

\(x^{8} + 102\) Copy content Toggle raw display

Invariants

Base field: $\Q_{17}$
Degree $d$: $8$
Ramification exponent $e$: $8$
Residue field degree $f$: $1$
Discriminant exponent $c$: $7$
Discriminant root field: $\Q_{17}(\sqrt{17\cdot 3})$
Root number: $-1$
$\card{ \Gal(K/\Q_{ 17 }) }$: $8$
This field is Galois and abelian over $\Q_{17}.$
Visible slopes:None

Intermediate fields

$\Q_{17}(\sqrt{17\cdot 3})$, 17.4.3.4

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: \( x^{8} + 102 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{7} + 8z^{6} + 11z^{5} + 5z^{4} + 2z^{3} + 5z^{2} + 11z + 8$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_8$ (as 8T1)
Inertia group:$C_8$ (as 8T1)
Wild inertia group:$C_1$
Unramified degree:$1$
Tame degree:$8$
Wild slopes:None
Galois mean slope:$7/8$
Galois splitting model:Not computed