Properties

Label 3.3.4.3
Base \(\Q_{3}\)
Degree \(3\)
e \(3\)
f \(1\)
c \(4\)
Galois group $C_3$

Related objects

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Defining polynomial

\( x^{3} - 3 x^{2} + 12 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $3$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $4$
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:$\Q_{3}$
Relative Eisenstein polynomial:\( x^{3} - 3 x^{2} + 12 \)

Invariants of the Galois closure

Galois group:$C_3$
Inertia group:$C_3$
Unramified degree:$1$
Tame degree:$1$
Wild slopes:[2]
Galois Mean Slope:$4/3$
Global Splitting Model:\( x^{3} - 21 x + 35 \)