In mathematics, the Conway polynomial C_p,n for the finite field F_pⁿ is a particular irreducible polynomial of degree n over F_p that can be used to define a standard representation of F_pⁿ 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,^[1] Macaulay2,^[2] Magma,^[3] SageMath,^[4] at the web site of Frank Lübeck,^[5] and at the Online Encyclopedia of Integer Sequences.^[6]

Background

Elements of F_pⁿ may be represented as sums of the form a_n−1βⁿ⁻¹ + … + a₁β + a₀ where β is a root of an irreducible polynomial of degree n over F_p 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 pⁿ up to isomorphism, the representation of the field elements depends on the choice of irreducible polynomial. The Conway polynomial is a way of standardizing this choice.

The non-zero elements of a finite field F form a cyclic group under multiplication, denoted F^*. A primitive element, α, of F_pⁿ is an element that generates F^*_pⁿ. 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_pⁿ (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 field F_pⁿ contains a unique subfield isomorphic to F_p^m for each m dividing n, and this accounts for all the subfields of F_pⁿ. For any m dividing n the cyclic group F^*_pⁿ contains a subgroup isomorphic to F^*_p^m. If α generates F^*_pⁿ, then the smallest power of α that generates this subgroup is α^r where r = (pⁿ − 1) / (p^m − 1). If f_n is a primitive polynomial for F_pⁿ with root α and 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_n(x) divide f_m(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.^[7]

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_dx^d − a_d−1x^d−1 + … + (−1)^da₀ (with terms alternately added and subtracted) and then expressed as the word a_d a_d−1 … a₀. 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.

Table

Conway polynomials C_p,n for the lowest values of p and n are tabulated below. All of these were first computed by Richard Parker and were taken from the tables of Frank Luebeck. The calculations can be verified using the basic methods of the next section with the assistance of algebra software.

Examples

To illustrate the definition, let us compute the first six Conway polynomials over F₅. By definition, a Conway polynomial is monic, primitive (which implies irreducible), and compatible with Conway polynomials of degree dividing its degree. The table below shows how imposing each of these conditions reduces the number of candidate polynomials.^[8]

Degree 1. The primitive elements of F₅ are 2 and 3. The two degree-1 polynomials with primitive roots are therefore x − 2 = x + 3 and x − 3 = x + 2, which correspond to the words 12 and 13, Since 12 is less than 13 in lexicographic ordering, C_5,1(x) = x + 3.

Degree 2. Since (5² − 1) / (5¹ − 1) = 6, compatibility requires that C_5,2 be chosen so that C_5,2(x) divides C_5,1(x⁶) = x⁶ + 3. The latter factorizes into three degree-2 polynomials, irreducible over F₅, namely x² + 2, x² + x + 2, and x² + 4x + 2. Of these x² + 2 is not primitive since it divides x⁸ − 1 implying that its roots have order at most 8, rather than the required 24. Both of the others are primitive and C_5,2 is chosen to be the lexicographically lesser of the two. Now x² + x + 2 = x² − 4x + 2 corresponds to the word 142 and x² + 4x + 2 = x² − x + 2 corresponds to the word 112, the latter being lexicographically less than the former. Hence C_5,2(x) = x² + 4x + 2.

Degree 3. Since (5³ − 1) / (5¹ − 1) = 31, compatibility requires that C_5,3(x) divide C_5,1(x³¹) = x³¹ + 3, which factorizes as a degree-1 polynomial times the product of ten primitive degree-3 polynomials. Of these, two have no quadratic term, x³ + 3x + 3 = x³ − 0x² + 3x − 2 and x³ + 4x + 3 = x³ − 0x² + 4x − 2, which correspond to the words 1032 and 1042. As 1032 is lexicographically less than 1042, C_5,3(x) = x³ + 3x + 3.

Degree 4. The proper divisors of 4 are 1 and 2. Compute (5⁴ − 1) / (5² − 1) = 26 and (5⁴ − 1) / (5¹ − 1) = 156, and note that 156 / 26 = (5² − 1) / (5² − 1) = 6, the same exponent as appeared in the compatibility condition for degree 2. In degree 4, compatibility requires that C_5,4 be chosen so that C_5,4(x) divides both C_5,2(x²⁶) = x⁵² + 4x²⁶ + 2 and C_5,1(x¹⁵⁶) = x¹⁵⁶ + 3. The second condition is redundant, however, because of the compatibility condition imposed when choosing C_5,2, which implies that C_5,2(x²⁶) divides C_5,1(x¹⁵⁶). In general, for composite degree d, the same reasoning implies that only the maximal proper divisors of d need be considered, that is, divisors of the form d / p, where p is a prime divisor of d. There are 13 factors of C_5,2(x²⁶), all of degree 4. All but one are primitive. Of the primitive ones, x⁴ + 4x² + 4x + 2 is lexicographically minimal.

Degree 5. The computation is similar to what was done in degrees 2 and 3: (5⁵ − 1) / (5¹ − 1) = 781; C_5,1(x⁷⁸¹) = x⁷⁸¹ + 3 has one factor of degree 1 and 156 factors of degree 5, of which 140 are primitive. The lexicographically least of the primitive factors is x⁵ + 4x + 3.

Degree 6. Taking into consideration the discussion above in connection with degree 4, the two compatibility conditions that need to be considered are that C_5,6(x) must divide C_5,2(x⁶⁵¹) = x¹³⁰² + 4x⁶⁵¹ + 2 and C_5,3(x¹²⁶) = x³⁷⁸ + 3x¹²⁶ + 3. It therefore must divide their greatest common divisor, x¹²⁶ + x¹⁰⁵ + 2x⁸⁴ + 3x⁴² + 2, which factorizes into 21 degree-6 polynomials, 18 of which are primitive. The lexicographically least of these is x⁶ + x⁴ + 4x³ + x² + 2.

Computation

Algorithms for computing Conway polynomials that are more efficient than brute-force search have been developed by Heath and Loehr.^[9] Lübeck indicates^[5] that their algorithm is a rediscovery of the method of Parker.

Notes

References

• Holt, Derek F.; Eick, Bettina; O'Brien, Eamonn A. (2005), Handbook of computational group theory, Discrete mathematics and its applications, vol. 24, CRC Press, ISBN 978-1-58488-372-2