Note: Solutions of a quartic GF(2) polynomial

This entry was posted on 2/1/2007 4:59 PM and is filed under Algebra.

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• 2/4/2007 9:51 PM Peter M. Maurer wrote:
  Sometimes it pays to take look back at the book before exerting a lot of effort. If p is any irreducible polynomial of degree m over GF(2), and a is any root of p, then GF(2^m) has basis vectors 1, a, a^2, ... , a^m-1. Furthermore, the complete list of roots of p is a, a^2, a^4, a^8, ... a^2^i, a^2^m-1. Thus, for every irreducible polynomial p over GF(2) a being a root of p also implies that a^2 is a root of p. Since the roots of p are mathematically indistinguishable, every square of every eigenvalue of a GF(2) matrix is also an eigenvalue of the same matrix.
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