Pure cubic number field
= Pure cubic number field
{title2=$K=\mathbb Q(\sqrt[3]{m})$}
A pure cubic number field is a degree-three <number field> generated by a root of $X^3-m$ for a rational number $m$ that is not a rational cube. For a positive <square-free integer> $m>1$, this polynomial is an <Eisenstein polynomial> at any prime dividing $m$. The <ring of integers of a number field> depends on congruences at $3$, as expressed by the <integral basis of a nonexceptional pure cubic field>.