Lemma 1

Let $P \in A[X]$ be a monic polynomial whose image $\overline{P}$ in $\kappa_A[X]$ is irreducible. Then the $A$-algebra $B = A[X]/(P)$ is local and finite over $A$, with maximal ideal $m_A B$, the canonical homomorphism $\rho : A \to B$ is local and flat, the residual extension $\kappa_A \to \kappa_B$ is algebraic and generated by the class $x$ of $X$, and the minimal polynomial of $x$ over $\kappa_A$ is $\overline{P}$.

Since the polynomial $P$ is monic, the $A$-module $B$ is free of finite type (A, IV, p. 10). The ring $B/m_A B$ is identified with $\kappa_A[X]/(\overline{P})$, and therefore is a field; hence the ideal $m_A B$ is maximal. Let $q$ be a maximal ideal of $B$; then the ideal $\rho^{-1}(q)$ is maximal (V, § 2, no 1, prop. 1); hence we have $\rho^{-1}(q) = m_A$, whence $q \supset m_A B$ and finally $q = m_A B$. Thus the ring $B$ is local. Lemma 1 follows at once.