Base \(\Q_{3}\)
Degree \(3\)
e \(3\)
f \(1\)
c \(3\)
Galois group $S_3$ (as 3T2)

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Defining polynomial

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


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: $-i$
$\card{ \Aut(K/\Q_{ 3 }) }$: $1$
This field is not Galois over $\Q_{3}.$
Visible slopes:$[3/2]$

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

Ramification polygon

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

Invariants of the Galois closure

Galois group:$S_3$ (as 3T2)
Inertia group:$S_3$ (as 3T2)
Wild inertia group:$C_3$
Unramified degree:$1$
Tame degree:$2$
Wild slopes:$[3/2]$
Galois mean slope:$7/6$
Galois splitting model: $x^{3} + 6 x + 3$ Copy content Toggle raw display