Properties

Label 3.3.3.2
Base \(\Q_{3}\)
Degree \(3\)
e \(3\)
f \(1\)
c \(3\)
Galois group $S_3$

Related objects

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Defining polynomial

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

Invariants

Base field: $\Q_{3}$
Degree $d$ : $3$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $3$
Discriminant root field: $\Q_{3}(\sqrt{3*})$
Root number: $i$
$|\Aut(K/\Q_{ 3 })|$: $1$
This field is not Galois 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:$\Q_{3}$
Relative Eisenstein polynomial:\( x^{3} + 3 x + 3 \)

Invariants of the Galois closure

Galois group:$S_3$
Inertia group:$S_3$
Unramified degree:$1$
Tame degree:$2$
Wild slopes:[3/2]
Galois Mean Slope:$7/6$
Global Splitting Model:\( x^{3} + 3 x + 3 \)