Properties

Label 5.8.4.1
Base \(\Q_{5}\)
Degree \(8\)
e \(2\)
f \(4\)
c \(4\)
Galois group $C_4\times C_2$ (as 8T2)

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Defining polynomial

\( x^{8} + 10 x^{6} + 125 x^{4} + 2500 \)

Invariants

Base field: $\Q_{5}$
Degree $d$ : $8$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $4$
Discriminant exponent $c$ : $4$
Discriminant root field: $\Q_{5}$
Root number: $-1$
$|\Gal(K/\Q_{ 5 })|$: $8$
This field is Galois and abelian over $\Q_{5}$.

Intermediate fields

$\Q_{5}(\sqrt{*})$, $\Q_{5}(\sqrt{5})$, $\Q_{5}(\sqrt{5*})$, 5.4.0.1, 5.4.2.1, 5.4.2.2

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

Unramified/totally ramified tower

Unramified subfield:5.4.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{4} + x^{2} - 2 x + 2 \)
Relative Eisenstein polynomial:$ x^{2} - 5 t^{2} \in\Q_{5}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2\times C_4$ (as 8T2)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$4$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{8} + 12 x^{6} + 30 x^{4} + 24 x^{2} + 4$