Properties

Label 3.3.0.1
Base \(\Q_{3}\)
Degree \(3\)
e \(1\)
f \(3\)
c \(0\)
Galois group $C_3$ (as 3T1)

Related objects

Learn more about

Defining polynomial

\( x^{3} - x + 1 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $3$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $3$
Discriminant exponent $c$ : $0$
Discriminant root field: $\Q_{3}$
Root number: $1$
$|\Gal(K/\Q_{ 3 })|$: $3$
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:3.3.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{3} - x + 1 \)
Relative Eisenstein polynomial:$ x - 3 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3$ (as 3T1)
Inertia group:Trivial
Unramified degree:$3$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{3} + x^{2} - 2 x - 1$