To construct a field with 16 elements, we begin by working with `ZZ_2` and aim to find an irreducible polynomial of degree 4. Consider the polynomial
`f(x) = x^4 + x + 1` in `ZZ_2[x]`.
Which of the following statements correctly justifies why f(x) is irreducible over `ZZ_2`:
A) f(x) is irreducible because it has no roots in `ZZ_2`, which means it cannot be factored into linear factors , and it is also not divisible by the only irreducible quadratic polynomial in `ZZ_2[x]`.
B) f(x) is irreducible because all of its coefficients are 1 , and any polynomial with all coefficients equal to 1 is automatically irreducible in `ZZ_2[x]` ?