Exercise

The polynomials f_i = _j a_ij X^j, i = 0, 1, ..., n - 1, form a basis for the vector space S = R[X]/(d) if and only if

gcd(f₀, f₁, ...,f_{n - 1}, d) = 1.

gcd(f₀, f₁, ...,f_{n - 1}) = 1.

The determinant of the matrix (a_ij) is nonzero.