Factorization of polynomials in Galois Field 2

Question

I want to know what is the general rule for factorization of the generator polynomial which is based on Galois Field, GF2?

Fact is, a generator of degree m must divide 1+x^n where remainder is zero. for example 1+x^7 ca be written as

(1+x+x3)(1+x2+x3)(1+x)

So either 1+x+x3 or 1+x2+x3 can be used.

What is the systematic rule for decomposing 1+x7?

Answer 1

A systematic way is to apply the Berlekamp algorithm. It gives $$ x^7+1=(x^3 + x^2 + 1)(x^3 + x + 1)(x + 1), $$ as you said. Or $$ x^{11}+1=(x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)(x + 1) $$