Properties

Label 5.10.10.2
Base \(\Q_{5}\)
Degree \(10\)
e \(5\)
f \(2\)
c \(10\)
Galois group $C_5^2 : C_8$ (as 10T18)

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Defining polynomial

\( x^{10} + 5 x^{6} + 10 x^{5} + 50 x^{2} + 25 x + 25 \)

Invariants

Base field: $\Q_{5}$
Degree $d$: $10$
Ramification exponent $e$: $5$
Residue field degree $f$: $2$
Discriminant exponent $c$: $10$
Discriminant root field: $\Q_{5}$
Root number: $1$
$|\Aut(K/\Q_{ 5 })|$: $1$
This field is not Galois over $\Q_{5}.$

Intermediate fields

$\Q_{5}(\sqrt{2})$

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{2} + 2 \)
Relative Eisenstein polynomial:$ x^{5} + \left(10 t + 15\right) x + 5 \in\Q_{5}(t)[x]$

Invariants of the Galois closure

Galois group:$C_5:D_5.C_4$ (as 10T18)
Inertia group:Intransitive group isomorphic to $C_5^2:C_4$
Unramified degree:$2$
Tame degree:$4$
Wild slopes:[5/4, 5/4]
Galois mean slope:$123/100$
Galois splitting model:$x^{10} + 60 x^{6} - 144 x^{5} - 1550 x^{2} + 8000 x - 10304$