Properties

Label 5.4.0.1
Base \(\Q_{5}\)
Degree \(4\)
e \(1\)
f \(4\)
c \(0\)
Galois group $C_4$ (as 4T1)

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Defining polynomial

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

Invariants

Base field: $\Q_{5}$
Degree $d$: $4$
Ramification exponent $e$: $1$
Residue field degree $f$: $4$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{5}(\sqrt{2})$
Root number: $1$
$\card{ \Gal(K/\Q_{ 5 }) }$: $4$
This field is Galois and abelian over $\Q_{5}.$
Visible slopes:None

Intermediate fields

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

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

Unramified/totally ramified tower

Unramified subfield:5.4.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{4} + 4 x^{2} + 4 x + 2 \) 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:$C_4$ (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:$x^{4} - 4 x^{2} + 2$