= Ramification groups of the splitting field of T3 minus 3T plus 3 over Q3
{title2=$G_0\cong S_3,\quad G_1\cong C_3,\quad G_i=1\ (i\geq2)$}
For the splitting field $M/\mathbb Q_3$ of $T^3-3T+3$, the cubic is <Eisenstein> and its discriminant is $-135$. The quadratic subfield is $\mathbb Q_3(\sqrt3)$, and the degree-six extension is totally ramified. If $\alpha$ is a root and $s=\sqrt3$, then $\pi=\alpha^2/s$ is a <uniformizer>. Transpositions have $v_M(\sigma\pi-\pi)=1$, while three-cycles have value $2$, giving the displayed <ramification groups>. Its <different exponent from ramification groups> is $7$.
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