Properties

Label 3.5.0.1
Base \(\Q_{3}\)
Degree \(5\)
e \(1\)
f \(5\)
c \(0\)
Galois group $C_5$

Related objects

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Defining polynomial

\( x^{5} - x + 1 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $5$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $5$
Discriminant exponent $c$ : $0$
Discriminant root field: $\Q_{3}$
Root number: $1$
$|\Gal(K/\Q_{ 3 })|$: $5$
This field is Galois and abelian over $\Q_{3}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 3 }$.

Unramified/totally ramified tower

Unramified subfield:3.5.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{5} - x + 1 \)
Relative Eisenstein polynomial:$ x - 3 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$C_5$
Inertia group:Trivial
Unramified degree:$5$
Tame degree:$1$
Wild slopes:None
Galois Mean Slope:$0$
Global Splitting Model:\( x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1 \)