A monic polynomial has leading coefficient one. An irreducible polynomial f over F_2 cannot be factored into two polynomials of smaller positive degree. In F_2[U]/(f), reduce every polynomial by division by f. If f has degree e, each element has a unique representative with powers 1,U,...,U^(e-1), and every nonzero element has a multiplicative inverse. There are 2^e such representatives. For f=U^2+U+1, U(U+1)=1 after reduction. U is an algebraic symbol, not a Boolean input: do not replace U^2 by U.