Background
Elements of F''p''''n'' may be represented as sums of the form ''a''''n''−1''β''''n''−1 + ... + ''a''1''β'' + ''a''0 where ''β'' is a root of an irreducible polynomial of degree ''n'' over Fp and the ''a''''j'' are elements of F''p''. Addition of field elements in this representation is simply vector addition. While there is a unique finite field of order ''p''''n'' up to isomorphism, the representation of the field elements depends on the choice of the irreducible polynomial. The Conway polynomial is a way of standardizing this choice. The non-zero elements of a finite field form a cyclic group under multiplication. A primitive element, ''α'', of F''p''''n'' is an element that generates this group. Representing the non-zero field elements as powers of ''α'' allows multiplication in the field to be performed efficiently. The primitive polynomial for ''α'' is the monic polynomial of smallest possible degree with coefficients in F''p'' that has ''α'' as a root in F''p''''n'' (the minimal polynomial for ''α''). It is necessarily irreducible. The Conway polynomial is chosen to be primitive, so that each of its roots generates the multiplicative group of the associated finite field. The subfields of F''p''''n'' are fields F''p''''m'' with ''m'' dividing ''n''. The cyclic group formed from the non-zero elements of F''p''''m'' is a subgroup of the cyclic group of F''p''''n''. If ''α'' generates the latter, then the smallest power of ''α'' that generates the former is ''α''''r'' where ''r'' = (''p''''n'' − 1)/(''p''''m'' − 1). If ''f''''n'' is a primitive polynomial for F''p''''n'' with root ''α'', and if ''f''''m'' is a primitive polynomial for F''p''''m'', then by Conway's definition, ''f''''m'' and ''f''''n'' are compatible if ''α''''r'' is a root of ''f''''m''. This necessitates that ''f''''m''(''x'') divide ''f''''n''(''x''''r''). This notion of compatibility is called norm-compatibility by some authors. The Conway polynomial for a finite field is chosen so as to be compatible with the Conway polynomials of each of its subfields. That it is possible to make the choice in this way was proved by Werner Nickel.Definition
The Conway polynomial ''C''''p'',''n'' is defined as the lexicographically minimal monic primitive polynomial of degree ''n'' over F''p'' that is compatible with ''C''''p'',''m'' for all ''m'' dividing ''n''. This is an inductive definition on ''n'': the base case is ''C''''p'',1(''x'') = ''x'' − ''α'' where ''α'' is the lexicographically minimal primitive element of F''p''. The notion of lexicographical ordering used is the following: * The elements of F''p'' are ordered 0 < 1 < 2 < ... < ''p'' − 1. * A polynomial of degree ''d'' in F''p'' 'x''is written ''a''''d''''x''''d'' − ''a''''d''−1''x''''d''−1 + ... + (−1)''d''''a''0 and then expressed as the word ''a''''d''''a''''d''−1 ... ''a''0. Two polynomials of degree ''d'' are ordered according to the lexicographical ordering of their corresponding words. Since there does not appear to be any natural mathematical criterion that would single out one monic primitive polynomial satisfying the compatibility conditions over all the others, the imposition of lexicographical ordering in the definition of the Conway polynomial should be regarded as a convention.Computation
Algorithms for computing Conway polynomials that are more efficient than brute-force search have been developed by Heath and Loehr. Lübeck indicates that their algorithm is a rediscovery of the method of Parker.Notes
References
*{{citation , title = Handbook of computational group theory , series = Discrete mathematics and its applications , volume = 24 , last1 = Holt , first1 = Derek F. , last2 = Eick , first2 = Bettina , last3 = O'Brien , first3 = Eamonn A. , publisher = CRC Press , year = 2005 , isbn = 978-1-58488-372-2 , url-access = registration , url = https://archive.org/details/handbookofcomput0000holt Finite fields Computer algebra John Horton Conway