Properties

Label 5.8.0.1
Base \(\Q_{5}\)
Degree \(8\)
e \(1\)
f \(8\)
c \(0\)
Galois group $C_8$ (as 8T1)

Related objects

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Defining polynomial

\(x^{8} + x^{2} - 2 x + 3\)  Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $8$
Ramification exponent $e$: $1$
Residue field degree $f$: $8$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{5}(\sqrt{2})$
Root number: $1$
$|\Gal(K/\Q_{ 5 })|$: $8$
This field is Galois and abelian over $\Q_{5}.$

Intermediate fields

$\Q_{5}(\sqrt{2})$, 5.4.0.1

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

Unramified/totally ramified tower

Unramified subfield:5.8.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{8} + x^{2} - 2 x + 3 \)  Toggle raw display
Relative Eisenstein polynomial:\( x - 5 \)$\ \in\Q_{5}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_8$ (as 8T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$8$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{8} - x^{7} - 7 x^{6} + 6 x^{5} + 15 x^{4} - 10 x^{3} - 10 x^{2} + 4 x + 1$