Base \(\Q_{3}\)
Degree \(3\)
e \(3\)
f \(1\)
c \(4\)
Galois group $C_3$ (as 3T1)

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Defining polynomial

\(x^{3} + 6 x^{2} + 12\) Copy content Toggle raw display


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: $1$
$\card{ \Gal(K/\Q_{ 3 }) }$: $3$
This field is Galois and abelian over $\Q_{3}.$
Visible slopes:$[2]$

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: \( x^{3} + 6 x^{2} + 12 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{2} + 2$
Associated inertia:$1$
Indices of inseparability:$[2, 0]$

Invariants of the Galois closure

Galois group:$C_3$ (as 3T1)
Inertia group:$C_3$ (as 3T1)
Wild inertia group:$C_3$
Unramified degree:$1$
Tame degree:$1$
Wild slopes:$[2]$
Galois mean slope:$4/3$
Galois splitting model: $x^{3} - 21 x + 35$ Copy content Toggle raw display