Properties

Label 5.10.5.1
Base \(\Q_{5}\)
Degree \(10\)
e \(2\)
f \(5\)
c \(5\)
Galois group $C_{10}$ (as 10T1)

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Defining polynomial

\(x^{10} - 50 x^{6} + 625 x^{2} - 12500\)  Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $10$
Ramification exponent $e$: $2$
Residue field degree $f$: $5$
Discriminant exponent $c$: $5$
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

$\Q_{5}(\sqrt{5})$, 5.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:5.5.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{5} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{2} - 5 \)$\ \in\Q_{5}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_{10}$ (as 10T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$5$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{10} - x^{9} - 13 x^{8} + 8 x^{7} + 46 x^{6} - 11 x^{5} - 52 x^{4} + 7 x^{3} + 18 x^{2} - 3 x - 1$