Properties

Label 89.4.0.1
Base \(\Q_{89}\)
Degree \(4\)
e \(1\)
f \(4\)
c \(0\)
Galois group $C_4$ (as 4T1)

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Defining polynomial

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

Invariants

Base field: $\Q_{89}$
Degree $d$: $4$
Ramification exponent $e$: $1$
Residue field degree $f$: $4$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{89}(\sqrt{3})$
Root number: $1$
$\card{ \Gal(K/\Q_{ 89 }) }$: $4$
This field is Galois and abelian over $\Q_{89}.$
Visible slopes:None

Intermediate fields

$\Q_{89}(\sqrt{3})$

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

Unramified/totally ramified tower

Unramified subfield:89.4.0.1 $\cong \Q_{89}(t)$ where $t$ is a root of \( x^{4} + 4 x^{2} + 72 x + 3 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x - 89 \) $\ \in\Q_{89}(t)[x]$ Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois group:$C_4$ (as 4T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$4$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{4} - x^{3} + 2 x^{2} + 4 x + 3$