Proposition 3

Let $A$ be a ring, $p$ a prime ideal of $A$ and $P \in A[T]$ a polynomial of Eisenstein for $p$. a) There does not exist a decomposition of the form $P = P_1 P_2$ where $P_1$ and $P_2$ are two monic polynomials of $A[T]$ distinct from 1. b) Suppose $A$ integrally closed, with field of fractions $K$. Then $P$ is irreducible in $K[T]$.

Let $\varphi$ be the canonical homomorphism of $A$ into the field of fractions $k$ of $A/p$ and let $\varphi': A[T] \to k[T]$ be the extension of $\varphi$ such that $\varphi'(T) = T$. Suppose that one has $P = P_1P_2$ where $P_1$ and $P_2$ are two monic polynomials of $A[T]$ distinct from 1. Then one has $T^d = \varphi'(P_1)\varphi'(P_2)$ in $k[T]$, denoting by $d$ the degree of $P$. If $d_i$ is the degree of $P_i$, one has therefore $\varphi'(P_i) = T^{d_i}$, that is to say $P_i(T) \equiv T^{d_i} \mod pA[T]$, and in particular $P_i(0) \in p$. But then $P(0) = P_1(0).P_2(0)$ belongs to $p^2$, contrary to the hypotheses. This proves a).

Assertion b) results from a) and prop. 11 of V, § 1, no 3.

Let $A$ be a noetherian local ring and $P_1, ..., P_r$ monic polynomials in $A[T]$, of degree $\geq 2$. Let $q$ be the ideal of $A[T_1, ..., T_r]$ generated by $P_1(T_1), ..., P_r(T_r)$ and let $B$ be the quotient $A$-algebra $A[T_1, ..., T_r]/q$. For $1 \leq i \leq r$, let $d_i$ denote the degree of $P_i$, $t_i$ the class of $T_i$ modulo $q$, and $\gamma_i$ the class of $c_i = P_i(0)$ modulo $m_A^2$. Suppose that $P_i(T) \equiv T^{d_i} \mod m_A A[T]$ for $1 \leq i \leq r$.