Conway polynomial (finite fields)

In mathematics, the Conway polynomial "C""p","n" for the finite field F"p""n" is a particular irreducible polynomial of degree "n" over F"p" that can be used to define a standard representation of F"p""n" as a splitting field of "C""p","n". Conway polynomials were named after John H. Conway by Richard A. Parker, who was the first to define them and compute examples. Conway polynomials satisfy a certain compatibility condition that had been proposed by Conway between the representation of a field and the representations of its subfields. They are important in computer algebra where they provide portability among different mathematical databases and computer algebra systems. Since Conway polynomials are expensive to compute, they must be stored to be used in practice. Databases of Conway polynomials are available in the computer algebra systems GAP, Macaulay2, Magma, SageMath, and at the web site of Frank Lübeck.
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:
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.