Conway polynomial (finite fields)

Template:Short description Template:About In mathematics, the Conway polynomial Template:Math for the finite field Template:Math is a particular irreducible polynomial of degree Template:Mvar over Template:Math that can be used to define a standard representation of Template:Math as a splitting field of Template:Math. 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, at the web site of Frank Lübeck, and at the Online Encyclopedia of Integer Sequences.

Background

Elements of ${𝐅}_{p^n}$ may be represented as sums of the form $a_{n-1}{\beta }^{n-1}+\cdots +a_1\beta +a_0$ where $\beta$ 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 $p^n$ 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 ${𝐅}^{\ast }$. A primitive element, $\alpha$, of ${𝐅}_{p^n}$ is an element that generates ${𝐅}_{p^n}^{\ast }$. Representing the non-zero field elements as powers of $\alpha$ allows multiplication in the field to be performed efficiently. The primitive polynomial for $\alpha$ is the monic polynomial of smallest possible degree with coefficients in ${𝐅}_p$ that has $\alpha$ as a root in ${𝐅}_{p^n}$ (the minimal polynomial for $\alpha$). 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 ${𝐅}_{p^n}$ contains a unique subfield isomorphic to ${𝐅}_{p^m}$ for each $m$ dividing $n$, and this accounts for all the subfields of ${𝐅}_{p^n}$. For any $m$ dividing $n$ the cyclic group ${𝐅}_{p^n}^{\ast }$ contains a subgroup isomorphic to ${𝐅}_{p^m}^{\ast }$. If $\alpha$ generates ${𝐅}_{p^n}^{\ast }$, then the smallest power of $\alpha$ that generates this subgroup is ${\alpha }^r$ where $r=(p^n-1)/(p^m-1)$. If $f_n$ is a primitive polynomial for ${𝐅}_{p^n}$ with root $\alpha$ and $f_m$ is a primitive polynomial for ${𝐅}_{p^m}$ then, by Conway's definition, $f_m$ and $f_n$ are compatible if ${\alpha }^r$ 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.

Definition

The Conway polynomial Template:Math is defined as the lexicographically minimal monic primitive polynomial of degree Template:Mvar over Template:Math that is compatible with Template:Math for all Template:Mvar dividing Template:Mvar. This is an inductive definition on Template:Mvar: the base case is Template:Math where Template:Mvar is the lexicographically minimal primitive element of Template:Math. The notion of lexicographical ordering used is the following:

• The elements of Template:Math are ordered Template:Math.
• A polynomial of degree Template:Mvar in Template:Math is written Template:Math (with terms alternately added and subtracted) and then expressed as the word Template:Math. 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 Template:Math for the lowest values of Template:Mvar and Template:Mvar 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.

Conway polynomials over Template:Math for small Template:Mvar and small degree.

Degree	Template:Math	Template:Math	Template:Math	Template:Math
1	Template:Math	Template:Math	Template:Math	Template:Math
2	Template:Math	Template:Math	Template:Math	Template:Math
3	Template:Math	Template:Math	Template:Math	Template:Math
4	Template:Math	Template:Math	Template:Math	Template:Math
5	Template:Math	Template:Math	Template:Math	Template:Math
6	Template:Math	Template:Math	Template:Math	Template:Math
7	Template:Math	Template:Math	Template:Math	Template:Math
8	Template:Math	Template:Math	Template:Math	Template:Math
9	Template:Math	Template:Math	Template:Math	Template:Math

Examples

To illustrate the definition, let us compute the first six Conway polynomials over Template:Math. 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.

Numbers of candidate polynomials over Template:Math as successive conditions are imposed.

Degree	monic	irreducible	primitive	compatible
1	5	5	2	2
2	25	10	4	2
3	125	40	20	10
4	625	150	48	12
5	3125	624	280	140
6	15625	2580	720	18

Degree 1. The primitive elements of Template:Math are 2 and 3. The two degree-1 polynomials with primitive roots are therefore Template:Math and Template:Math, which correspond to the words 12 and 13, Since 12 is less than 13 in lexicographic ordering, Template:Math.

Degree 2. Since Template:Math, compatibility requires that Template:Math be chosen so that Template:Math divides Template:Math. The latter factorizes into three degree-2 polynomials, irreducible over Template:Math, namely Template:Math, Template:Math, and Template:Math. Of these Template:Math is not primitive since it divides Template:Math implying that its roots have order at most 8, rather than the required 24. Both of the others are primitive and Template:Math is chosen to be the lexicographically lesser of the two. Now Template:Math corresponds to the word 142 and Template:Math corresponds to the word 112, the latter being lexicographically less than the former. Hence Template:Math.

Degree 3. Since Template:Math, compatibility requires that Template:Math divide Template:Math, which factorizes as a degree-1 polynomial times the product of ten primitive degree-3 polynomials. Of these, two have no quadratic term, Template:Math and Template:Math, which correspond to the words 1032 and 1042. As 1032 is lexicographically less than 1042, Template:Math.

Degree 4. The proper divisors of Template:Math are Template:Math and Template:Math. Compute Template:Math and Template:Math, and note that Template:Math, the same exponent as appeared in the compatibility condition for degree 2. In degree 4, compatibility requires that Template:Math be chosen so that Template:Math divides both Template:Math and Template:Math. The second condition is redundant, however, because of the compatibility condition imposed when choosing Template:Math, which implies that Template:Math divides Template:Math. In general, for composite degree Template:Mvar, the same reasoning implies that only the maximal proper divisors of Template:Mvar need be considered, that is, divisors of the form Template:Math, where Template:Mvar is a prime divisor of Template:Mvar. There are 13 factors of Template:Math, all of degree 4. All but one are primitive. Of the primitive ones, Template:Math is lexicographically minimal.

Degree 5. The computation is similar to what was done in degrees 2 and 3: Template:Math; Template:Math 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 Template:Math.

Degree 6. Taking into consideration the discussion above in connection with degree 4, the two compatibility conditions that need to be considered are that Template:Math must divide Template:Math and Template:Math. It therefore must divide their greatest common divisor, Template:Math, which factorizes into 21 degree-6 polynomials, 18 of which are primitive. The lexicographically least of these is Template:Math.

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.

References

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