Properties

Label 5.5.0.1
Base \(\Q_{5}\)
Degree \(5\)
e \(1\)
f \(5\)
c \(0\)
Galois group $C_5$ (as 5T1)

Related objects

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Defining polynomial

\( x^{5} - x + 2 \)

Invariants

Base field: $\Q_{5}$
Degree $d$ : $5$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $5$
Discriminant exponent $c$ : $0$
Discriminant root field: $\Q_{5}$
Root number: $1$
$|\Gal(K/\Q_{ 5 })|$: $5$
This field is Galois and abelian over $\Q_{5}$.

Intermediate fields

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

Unramified/totally ramified tower

Unramified subfield:5.5.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{5} - x + 2 \)
Relative Eisenstein polynomial:$ x - 5 \in\Q_{5}(t)[x]$

Invariants of the Galois closure

Galois group:$C_5$ (as 5T1)
Inertia group:Trivial
Unramified degree:$5$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$