A Note on the Construction of Irreducible Polynomials

Polynomials as Lists

For internal computations, polynomials are maintained as lists, such that the zero order coefficient corresponds to the first list element.

The Polynomial

To calculate sufficiently many irreducible polynomials we use a well-known fact about polynomials over finite fields: For every finte field and for all ,

,

where the product is extended over all monic irreducible polynomials for which divides . Selecting to contain the proper prime factors we get the disired list of monic pairwise irreducible polynomials.
Mathematica supports factoring polynomials over prime field, so the above approach works fine for prime. For example to get all the monic irreducible polynomials of degree 1,2,3 in factor in :

1	"x"
0		""	""
		""	""
		""	""
	0		""
			""
			""
		0
		0
	0

Converted by Mathematica November 18, 1998