Properties

Label 5.8.6.1
Base \(\Q_{5}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(6\)
Galois group $C_4\times C_2$ (as 8T2)

Related objects

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Defining polynomial

\( x^{8} - 5 x^{4} + 400 \)

Invariants

Base field: $\Q_{5}$
Degree $d$ : $8$
Ramification exponent $e$ : $4$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $6$
Discriminant root field: $\Q_{5}$
Root number: $-1$
$|\Gal(K/\Q_{ 5 })|$: $8$
This field is Galois and abelian over $\Q_{5}$.

Intermediate fields

$\Q_{5}(\sqrt{*})$, $\Q_{5}(\sqrt{5})$, $\Q_{5}(\sqrt{5*})$, 5.4.2.1, 5.4.3.2, 5.4.3.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{5}(\sqrt{*})$ $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)
Relative Eisenstein polynomial:$ x^{4} - 5 t^{4} \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 $C_4$
Unramified degree:$2$
Tame degree:$4$
Wild slopes:None
Galois mean slope:$3/4$
Galois splitting model:\( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \)