Conway polynomial (finite fields)

In mathematics, the Conway polynomial $C_{p,n}$ for the finite field $F_{p^{n}}|undefined$ is a particular irreducible polynomial of degree $n$ over $F_{p}$ that can be used to define a standard representation of $F_{p^{n}}|undefined$ 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, at the web site of Frank Lübeck, and at the Online Encyclopedia of Integer Sequences.

Background
Elements of $C_{p,n}$ may be represented as sums of the form $p = 2$ where $β$ is a root of an irreducible polynomial of degree $n$ over $F_{p^{n}}|undefined$ and the $a_{n−1}β^{n−1} + … + a_{1}β + a_{0}$ 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 $a_{j}$ 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 $F_{p}$ form a cyclic group under multiplication, denoted $p^{n}$. A primitive element, $α$, of $F$ is an element that generates $F^{*}$. 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 $F_{p^{n}}|undefined$ that has $α$ as a root in $F^{*}_{p^{n}}|undefined$ (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 $F_{p}$ contains a unique subfield isomorphic to $F_{p^{n}}|undefined$ for each $m$ dividing $n$, and this accounts for all the subfields of $F_{p^{n}}|undefined$. For any $m$ dividing $n$ the cyclic group $F_{p^{m}}|undefined$ contains a subgroup isomorphic to $F_{p^{n}}|undefined$. If $α$ generates $F^{*}_{p^{n}}|undefined$, then the smallest power of $α$ that generates this subgroup is $F^{*}_{p^{m}}|undefined$ where $F^{*}_{p^{n}}|undefined$. If $α^{r}$ is a primitive polynomial for $r = (p^{n} − 1) / (p^{m} − 1)$ with root $α$ and $f_{n}$ is a primitive polynomial for $F_{p^{n}}|undefined$ then, by Conway's definition, $f_{m}$ and $F_{p^{m}}|undefined$ are compatible if $f_{m}$ is a root of $f_{n}$. This necessitates that $α^{r}$ divide $f_{m}$. 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 $f_{n}(x)$ is defined as the lexicographically minimal monic primitive polynomial of degree $n$ over $f_{m}(x^{r})$ that is compatible with $C_{p,n}$ for all $m$ dividing $n$. This is an inductive definition on $n$: the base case is $F_{p}$ where $α$ is the lexicographically minimal primitive element of $C_{p,m}$. 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.
 * The elements of $C_{p,1}(x) = x − α$ are ordered $F_{p}$.
 * A polynomial of degree $d$ in $F_{p}$ is written $0 < 1 < 2 < … < p − 1$ (with terms alternately added and subtracted) and then expressed as the word $F_{p}[x]$. Two polynomials of degree d are ordered according to the lexicographical ordering of their corresponding words.

Table
Conway polynomials $a_{d}x^{d} − a_{d−1}x^{d−1} + … + (−1)^{d}a_{0}$ 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 $a_{d} a_{d−1} … a_{0}$. 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.

Degree 1. The primitive elements of $C_{p,n}$ are 2 and 3. The two degree-1 polynomials with primitive roots are therefore $F_{p}$ and $F_{2}$, which correspond to the words 12 and 13, Since 12 is less than 13 in lexicographic ordering, $F_{3}$.

Degree 2. Since $F_{5}$, compatibility requires that $F_{7}$ be chosen so that $x + 1$ divides $x + 1$. The latter factorizes into three degree-2 polynomials, irreducible over $x + 3$, namely $x + 4$, $x^{2} + x + 1$, and $x^{2} + 2x + 2$. Of these $x^{2} + 4x + 2$ is not primitive since it divides $x^{2} + 6x + 3$ implying that its roots have order at most 8, rather than the required 24. Both of the others are primitive and $x^{3} + x + 1$ is chosen to be the lexicographically lesser of the two. Now $x^{3} + 2x + 1$ corresponds to the word 142 and $x^{3} + 3x + 3$ corresponds to the word 112, the latter being lexicographically less than the former. Hence $x^{3} + 6x^{2} + 4$.

Degree 3. Since $x^{4} + x + 1$, compatibility requires that $x^{4} + 2x^{3} + 2$ divide $x^{4} + 4x^{2} + 4x + 2$, which factorizes as a degree-1 polynomial times the product of ten primitive degree-3 polynomials. Of these, two have no quadratic term, $x^{4} + 5x^{2} + 4x + 3$ and $x^{5} + x^{2} + 1$, which correspond to the words 1032 and 1042. As 1032 is lexicographically less than 1042, $x^{5} + 2x + 1$.

Degree 4. The proper divisors of $x^{5} + 4x + 3$ are $x^{5} + x + 4$ and $x^{6} + x^{4} + x^{3} + x + 1$. Compute $x^{6} + 2x^{4} + x^{2} + 2x + 2$ and $x^{6} + x^{4} + 4x^{3} + x^{2} + 2$, and note that $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3$, the same exponent as appeared in the compatibility condition for degree 2. In degree 4, compatibility requires that $x^{7} + x + 1$ be chosen so that $x^{7} + 2x^{2} + 1$ divides both $x^{7} + 3x + 3$ and $x^{7} + 6x + 4$. The second condition is redundant, however, because of the compatibility condition imposed when choosing $x^{8} + x^{4} + x^{3} + x^{2} + 1$, which implies that $x^{8} + 2x^{5} + x^{4} + 2x^{2} + 2x + 2$ divides $x^{8} + x^{4} + 3x^{2} + 4x + 2$. In general, for composite degree $p$, the same reasoning implies that only the maximal proper divisors of $d$ need be considered, that is, divisors of the form $x^{8} + 4x^{3} + 6x^{2} + 2x + 3$, where $d$ is a prime divisor of $d$. There are 13 factors of $x^{9} + x^{4} + 1$, all of degree 4. All but one are primitive. Of the primitive ones, $x^{9} + 2x^{3} + 2x^{2} + x + 1$ is lexicographically minimal.

Degree 5. The computation is similar to what was done in degrees 2 and 3: $x^{9} + 2x^{3} + x + 3$; $x^{9} + 6x^{4} + x^{3} + 6x + 4$ 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 $F_{5}$.

Degree 6. Taking into consideration the discussion above in connection with degree 4, the two compatibility conditions that need to be considered are that $5^{d}$ must divide $F_{5}$ and $F_{5}$. It therefore must divide their greatest common divisor, $F_{5}$, which factorizes into 21 degree-6 polynomials, 18 of which are primitive. The lexicographically least of these is $F_{5}$.

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.