Papers › Sorting and labelling integral ideals in a number field
Sorting and labelling integral ideals in a number field
John Cremona, Aurel Page, Andrew V. Sutherland
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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.
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