Final answer:
The gcd of the polynomials (x³ + 1) and (x² + x + 1) over GF(2) is (x² + x + 1) as it is a factor of (x³ + 1) and irreducible over this field.
Step-by-step explanation:
The greatest common divisor (gcd) of two polynomials over GF(2) can be found using polynomial division or by factoring, similar to the process used with integers. In the field GF(2), each coefficient is considered modulo 2, which means we only deal with the coefficients 0 and 1. In this particular case, we need to determine the gcd of the polynomials (x³ + 1) and (x² + x + 1) over GF(2).
Firstly, let's notice that (x³ + 1) can be factored over GF(2) as (x + 1)(x² + x + 1) since 1 is the only element in the field GF(2) and x² + x + 1 is irreducible over GF(2).
Therefore, the polynomial (x² + x + 1) is a factor of both, which makes it the gcd of (x³ + 1) and (x² + x + 1).