Properties

Label 5.10.10.13
Base \(\Q_{5}\)
Degree \(10\)
e \(5\)
f \(2\)
c \(10\)
Galois group $(C_5^2 : C_4) : C_2$ (as 10T17)

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Defining polynomial

\( x^{10} + 15 x^{6} + 10 x^{5} + 100 x^{2} + 75 x + 25 \)

Invariants

Base field: $\Q_{5}$
Degree $d$: $10$
Ramification exponent $e$: $5$
Residue field degree $f$: $2$
Discriminant exponent $c$: $10$
Discriminant root field: $\Q_{5}(\sqrt{2})$
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 + 20\right) x + 5 \in\Q_{5}(t)[x]$

Invariants of the Galois closure

Galois group:$D_5:F_5$ (as 10T17)
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:$x^{10} - 10 x^{8} + 35 x^{6} - 4 x^{5} - 50 x^{4} + 20 x^{3} + 25 x^{2} - 20 x - 28$