Exercise 15, § 3

Let K be a complete field for a discrete valuation $v$, A the ring of the valuation, k its residue field, P a polynomial of $A[X_1, \ldots, X_n]$ of total degree d having the following property: there exists an algebraic extension $K'$ of K such that in $K'[X_1, \ldots, X_n]$, P is a product of polynomials of total degree 1. Suppose moreover that there exist two polynomials Q, R of $A[X_1, \ldots, X_n]$ such that Q is of total degree s and contains a monomial $aX_1^s$ where $\varphi(a) \neq 0$ ($\varphi$ denoting the canonical homomorphism $A \to k$), that R is of degree $\leq d - s$, and that one has $\bar{P} = \bar{Q} \cdot \bar{R}$ (notations of Chapter III, § 4). Now prove that there then exist in $A[X_1, \ldots, X_n]$ two polynomials $Q_0, R_0$ of respective degrees s, $d - s$, such that $P = Q_0 R_0$, $\bar{Q} = \bar{Q}_0, \bar{R} = \bar{R}_0$ and that $Q_0$ contains a monomial $a_0 X_1^s$ with $\varphi(a) = \varphi(a_0)$, (Consider P, Q, R as polynomials in $X_1$ with coefficients in the ring B, the completion of $A[X_2, \ldots, X_n]$ for the valuation obtained by extending $v$ according to the method of Chapter VI, § 10, No. 1, Prop. 2; then apply Hensel's lemma; finally use the initial assumption on P).