Properties

Label 17.10.5.2
Base \(\Q_{17}\)
Degree \(10\)
e \(2\)
f \(5\)
c \(5\)
Galois group $C_{10}$ (as 10T1)

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Defining polynomial

\(x^{10} - 83521 x^{2} + 8519142\)  Toggle raw display

Invariants

Base field: $\Q_{17}$
Degree $d$: $10$
Ramification exponent $e$: $2$
Residue field degree $f$: $5$
Discriminant exponent $c$: $5$
Discriminant root field: $\Q_{17}(\sqrt{17\cdot 3})$
Root number: $-1$
$|\Gal(K/\Q_{ 17 })|$: $10$
This field is Galois and abelian over $\Q_{17}.$

Intermediate fields

$\Q_{17}(\sqrt{17\cdot 3})$, 17.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:17.5.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{5} - x + 6 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{2} - 17 t \)$\ \in\Q_{17}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_{10}$ (as 10T1)
Inertia group:Intransitive group isomorphic to $C_2$
Wild inertia group:$C_1$
Unramified degree:$5$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:Not computed