Properties

Label 5.10.17.5
Base \(\Q_{5}\)
Degree \(10\)
e \(10\)
f \(1\)
c \(17\)
Galois group $C_{10}$ (as 10T1)

Related objects

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Defining polynomial

\(x^{10} - 5 x^{8} + 55\)  Toggle raw display

Invariants

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

Intermediate fields

$\Q_{5}(\sqrt{5})$, 5.5.8.4

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{5}$
Relative Eisenstein polynomial:\( x^{10} - 30 x^{8} + 225 x^{6} + 1805 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_{10}$ (as 10T1)
Inertia group:$C_{10}$
Unramified degree:$1$
Tame degree:$2$
Wild slopes:[2]
Galois mean slope:$17/10$
Galois splitting model:$x^{10} - 110 x^{8} + 4235 x^{6} - 1276 x^{5} - 66550 x^{4} + 70180 x^{3} + 366025 x^{2} - 771980 x + 404624$