It should be remarked that, thanks to the notion of reducedness, our algorithm
gives for every cubic field a canonical equation, which we can call
reduced, and which in addition has all the
nice properties described in Section 4. In particular the integral basis
and decomposition of primes is immediate. One consequence is that, when the
cubic number field does not have a power basis, the equation that we will
find will not be monic. On the other hand, if there exists a power basis,
the reduced equation produced by our algorithm is not necessarily monic.
We give a few examples.