Random polynomials over finite fields

Flip a coin to select a random polynomial over F2. The sequence HTHHHTH, for example, corresponds to the sequence 1011101 or to the polynomial 1 + x2 + x3 + x4 + x6. Do it again to get another random polynomial. What are the chances that the two polynomials are coprime? We present some data. Consider sequences of length 3, which means polynomials of degree 2 or less. There are 64 pairs of such polynomials. Two polynomials f and g are coprime if and only if their greatest common divisor is 1, and we dene the gcd of f and g to be the unique monic polynomial that generates the ideal generated by f and g. (Over F2 it is unnecessary to specify ‘monic.’). Thus, over F2 gcd(0; f) = f and so 0 is coprime only to 1 and to no other polynomial. In general the gcd of 0 and f is the unique monic polynomial that is a scalar multiple of f, and 0 is coprime to the non-zero constant polynomials. In the table below an asterisk means the corresponding pair is coprime. 0 1 x 1 + x x2 1 + x2 x+ x2 1 + x+ x