  We define a scheme for labelling and ordering integral ideals of number
fields, including prime ideals as a special case. The order we define depends
only on the choice of a monic irreducible integral defining polynomial for each
field $K$, and we start by defining for each field its unique reduced defining
polynomial, after Belabas. We define a total order on the set of prime ideals
of $K$ and then extend this to a total order on the set of all nonzero integral
ideals of $K$. This order allows us to give a unique label of the form $N.i$,
where $N$ is its norm and $i$ is the index of the ideal in the ordered list of
all ideals of norm $N$. Our ideal labelling scheme has several nice properties:
for a given norm, prime ideals always appear first, and given the factorisation
of the norm, the bijection between ideals of norm $N$ and labels is computable
in polynomial time.
  Our motivation for this is to have a well-defined and concise way to sort and
label ideals for use in databases such as the LMFDB. We have implemented
algorithms which realise this scheme, in Sage, Magma and Pari.
