From: Daniel Reischert <daniel@informatik.uni-bonn.de>
Newsgroups: sci.math.symbolic
Subject: Re: Large irreducible polynomials
Date: 22 Mar 1996 16:42:49 GMT

In [1] you can find the construction of 
irreducible polynomials of degree m*t out of 
irreducible polynomials of degree m, if t is chosen appropriately.
Taking 
	f1(x) = x^4 + x   + 1  and
	f2(x) = x^4 + x^3 + 1, which are irreducible over GF(2),
you can construct (choosing t= 3^3 * 5^4 = 16.875) 
	g1(x) = x^67.500 + x^16.875 + 1 and
	g2(x) = x^67.500 + x^50.625 + 1,
which are also irreducible, by the theorem in [1].

These polynomials seem to be quite sparse ...
but I think, you can avoid this by switching to
g1(x+1) and g2(x+1). Right?

Another "type" of constrution (it might be the
same, I didn't check this) can be found in [2]. 
The polynomials constructed in this paper are
of degree 2^n, but only one polynomial per
degree is gained. Maybe there are some variations
of this procedure?

Daniel Reischert, Bonn, Germany, daniel@cs.bonn.edu

[1] R. Lidl, H. Niederreiter, "Introduction to Finite Fields and
	Their Applications".
	Encyklopedia of Mathematic and its Applications, 
	Section Algebra, Vol.20, Addison-Wesley, 
	Reading, Massachusetts (1983), pp. 88--89
[2] D. G. Cantor, "On Arithmetical Algorithms over Finite Fields",
	Journal of Combinatorial Theory A 50 (1989), 285--300.