Properties

Label 5.1.0.1
Base \(\Q_{5}\)
Degree \(1\)
e \(1\)
f \(1\)
c \(0\)
Galois group Trivial (as 1T1)

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Defining polynomial

\(x + 3\) Copy content Toggle raw display

Invariants

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

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 5 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{5}$
Relative Eisenstein polynomial: \( x - 5 \) Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois group: $C_1$ (as 1T1)
Inertia group: $C_1$ (as 1T1)
Wild inertia group: $C_1$
Unramified degree: $1$
Tame degree: $1$
Wild slopes: None
Galois mean slope: $0$
Galois splitting model: $x + 2$ Copy content Toggle raw display