$GL_n(\mathbb{C})$-conjugation invariant $\mathbb{C}$-valued polynomial has certain form.

Question

What is the easiest way to see that any $GL_n(\mathbb{C})$-conjugation invariant $\mathbb{C}$-valued polynomial function on $M_n(\mathbb{C})$ has the form $f(A) = F(p_0(A), \dots, p_{n-1}(A))$ for some $F \in \mathbb{C}[x_0, x_1, \dots, x_{n-1}]$, where the functions $p_j$ are defined by the identity$$p_0(A) + tp_1(A) + \dots + t^{n-1}p_{n-1}(A) + t^n = \det(tI - A)?$$