Lemma 5

Let t be an element of A. There exists a commutative ring C containing A as subring, a subring B of C containing A, an element $\tau$ of B invertible in C and a ring homomorphism $p : B \to A$ such that $p(\tau) = t$ and the restriction of p to A is equal to Id.

It suffices to take for B the algebra $A^{(\mathbf{N})}$ of the monoid $\mathbf{N}$, that is, the polynomial algebra $A[\tau]$ in one indeterminate $\tau$, for C the algebra $A^{(\mathbf{Z})}$ of the group $\mathbf{Z}$ and for p the homomorphism $P \mapsto P(t)$ of $A(\tau)$ into A.

With the notation of Lemma 5, where we have taken t = t_p, let us put

in B[X]. The coefficient of $X^p$ in F is invertible in C; if we consider F, g, $g_1$ as polynomials of C[X] we thus have by what has been said. The resultants are unchanged if we consider F, g and $g_1$ as polynomials in B[X]. Since $^\rho F = f$, $^\rho g = g$, $^\rho g_1 = g_1$, formula (35) follows from (38) and Remark 2 (IV, p. 77).