Integral Basis for Cubic Fields

Integral basis for ${\bf Q}(\root3\of a)$ is given in Theorem 7.3.2 of Alaca and Williams, Introductory Algebraic Number Theory:

Let $m$ be a cubefree integer. Set $m=hk^2$, where $h$ is squarefree, so that $k$ is squarefree and $(h,k)=1$. Set $\theta=m^{1/3}$ and $K={\bf Q}(\theta)$. Then an integral basis for $K$ is $$\eqalign{&\{{1,\theta,\theta^2/k\}},{\rm\ if\ }m^2\not\equiv1\pmod9,\cr&\{{1,\theta,(k^2\pm k^2\theta+\theta^2)/3k\}},{\rm\ if\ }m\equiv\pm1\pmod9.\cr}$$

You can find an "elementary" proof in example 4.3.6 of Murty and Esmonde, Problems in Algebraic Number Theory, here.