Properties

Label 17.10.9.2
Base \(\Q_{17}\)
Degree \(10\)
e \(10\)
f \(1\)
c \(9\)
Galois group $F_{5}\times C_2$ (as 10T5)

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Defining polynomial

\(x^{10} + 51\)  Toggle raw display

Invariants

Base field: $\Q_{17}$
Degree $d$: $10$
Ramification exponent $e$: $10$
Residue field degree $f$: $1$
Discriminant exponent $c$: $9$
Discriminant root field: $\Q_{17}(\sqrt{17\cdot 3})$
Root number: $1$
$|\Aut(K/\Q_{ 17 })|$: $2$
This field is not Galois over $\Q_{17}.$

Intermediate fields

$\Q_{17}(\sqrt{17\cdot 3})$, 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:\( x^{10} + 51 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2\times F_5$ (as 10T5)
Inertia group:$C_{10}$
Wild inertia group:$C_1$
Unramified degree:$4$
Tame degree:$10$
Wild slopes:None
Galois mean slope:$9/10$
Galois splitting model:$x^{10} + 51$  Toggle raw display