Properties

Label 131.8.0.1
Base \(\Q_{131}\)
Degree \(8\)
e \(1\)
f \(8\)
c \(0\)
Galois group $C_8$ (as 8T1)

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Defining polynomial

\(x^{8} + 3 x^{4} + 72 x^{3} + 116 x^{2} + 104 x + 2\) Copy content Toggle raw display

Invariants

Base field: $\Q_{131}$
Degree $d$: $8$
Ramification exponent $e$: $1$
Residue field degree $f$: $8$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{131}(\sqrt{2})$
Root number: $1$
$\card{ \Gal(K/\Q_{ 131 }) }$: $8$
This field is Galois and abelian over $\Q_{131}.$
Visible slopes:None

Intermediate fields

$\Q_{131}(\sqrt{2})$, 131.4.0.1

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

Unramified/totally ramified tower

Unramified subfield:131.8.0.1 $\cong \Q_{131}(t)$ where $t$ is a root of \( x^{8} + 3 x^{4} + 72 x^{3} + 116 x^{2} + 104 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x - 131 \) $\ \in\Q_{131}(t)[x]$ Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

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:$x^{8} - x^{7} - 7 x^{6} + 6 x^{5} + 15 x^{4} - 10 x^{3} - 10 x^{2} + 4 x + 1$