Corollary 3

If A is factorial (VII, § 3, No. 3, Prop. 2), the prime ideals of height 1 of A are the principal ideals generated by the extremal elements of A. If moreover A is local, one has $\dim(A/p) = \dim(A) - 1$ for every prime ideal $p$ of height 1 of A.

Let x be an extremal element of A. Then Ax is a prime ideal because x is extremal, of height 1 because A is an integral domain (No. 1, Prop. 1). Let p be a prime ideal of height 1 of A. Then $V(p)$ is an irreducible component of a hypersurface $V(Ax)$ for a suitable x (Prop. 4, c)). Let $x = \prod_i y_i^{n_i}$ be a decomposition of x into a product of extremal elements such that $y_i$ and $y_j$ are relatively prime if $i \neq j$. The irreducible components of $V(Ax)$ are the $V(Ay_i)$. Hence $p = Ay_i$ for a suitable i. The last assertion follows from Corollary 2 to Prop. 2 of No. 1.