= Ramification of the splitting field of X3 plus pX plus p
{title2=$X^3+pX+p$}
For an odd prime $p$, this <Eisenstein polynomial> generates a totally ramified cubic root field. Its <discriminant> is $-p^2(4p+27)$. For $p>3$ the unit part is a square exactly when $p\equiv1\pmod3$. The <splitting field> then has cyclic <Galois group> of order three; otherwise it has <symmetric group> $S_3$, with an <unramified extension> of degree two and <inertia group> $C_3$. Both cases are <tamely ramified>, so all positive <ramification groups> are trivial.
At $p=3$, adjoining $\beta=\sqrt{-39}$ gives a ramified quadratic subfield, and the <splitting field> has total <ramification index> six. A root $\alpha$ has normalized <valuation> two, while $\beta$ has <valuation> three, so $\pi=\beta/\alpha$ is a <uniformizer>. The derivative $3\alpha^2+3$ has <valuation> six, and all distinct root differences have <valuation> three. A three-cycle fixes $\beta$, giving $v_L(\sigma\pi-\pi)=2$. A transposition fixing $\alpha$ negates $\beta$, giving <valuation> one; conjugation handles the other transpositions. Hence $G_0=S_3$, $G_1=C_3$, and $G_i=1$ for $i\ge2$. The <different exponent> is seven.
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