= Integral basis of a cubefree pure cubic field
{title2=$1,\alpha,\alpha^2/m_2,\quad\alpha^3=m_1m_2^2$}
Let $m_1,m_2$ be coprime positive <square-free integers>, with $m_1\not\equiv\pm m_2\pmod9$. The displayed <integral basis> has <discriminant> $-27m_1^2m_2^2$. Both nonconstant generators are integral, and their products are integral linear combinations of the three basis elements. At a prime dividing $m_1$, the cubic for $\alpha$ is <Eisenstein>; at a prime dividing $m_2$, the cubic for $\alpha^2/m_2$ is <Eisenstein>. If $3\nmid m_1m_2$, then $m_1m_2^2\not\equiv\pm1\pmod9$, and a shift by $1$ or $-1$ gives an <Eisenstein polynomial> at three. These local <uniformizer> generators show that the order is maximal at all primes dividing its <discriminant>; elsewhere the <discriminant-index formula for an integral lattice> already gives maximality.
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