Properties

Label 5.10.10.7
Base \(\Q_{5}\)
Degree \(10\)
e \(5\)
f \(2\)
c \(10\)
Galois group $F_{5}\times C_2$ (as 10T5)

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Defining polynomial

\( x^{10} + 10 x^{8} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 12 \)

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: $-1$
$|\Aut(K/\Q_{ 5 })|$: $2$
This field is not Galois over $\Q_{5}.$

Intermediate fields

$\Q_{5}(\sqrt{2})$, 5.5.5.2

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} + 5 x + 5 \in\Q_{5}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2\times F_5$ (as 10T5)
Inertia group:Intransitive group isomorphic to $F_5$
Unramified degree:$2$
Tame degree:$4$
Wild slopes:[5/4]
Galois mean slope:$23/20$
Galois splitting model:$x^{10} - 3 x^{5} + 3$