Properties

Label 5.9.0.1
Base \(\Q_{5}\)
Degree \(9\)
e \(1\)
f \(9\)
c \(0\)
Galois group $C_9$ (as 9T1)

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Defining polynomial

\(x^{9} + 2 x^{3} + x + 3\) Copy content Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $9$
Ramification exponent $e$: $1$
Residue field degree $f$: $9$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{5}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 5 }) }$: $9$
This field is Galois and abelian over $\Q_{5}.$
Visible slopes:None

Intermediate fields

5.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:5.9.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{9} + 2 x^{3} + x + 3 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x - 5 \) $\ \in\Q_{5}(t)[x]$ Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois group: $C_9$ (as 9T1)
Inertia group: trivial
Wild inertia group: $C_1$
Unramified degree: $9$
Tame degree: $1$
Wild slopes: None
Galois mean slope: $0$
Galois splitting model:$x^{9} - x^{8} - 8 x^{7} + 7 x^{6} + 21 x^{5} - 15 x^{4} - 20 x^{3} + 10 x^{2} + 5 x - 1$