Exercise 15, § 3

Let K be a field which is complete with respect to the valuation $v$, A the ring of the valuation, $k$ its residue field and P a polynomial in $A[X_1, \ldots, X_n]$ of total degree d with 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 further that there exist two polynomials Q, R in $A[X_1, \ldots, X_n]$ such that Q is of total degree s and contains a monomial $aX_1^s$ where $\phi(a) \neq 0$ ($\phi$ denoting the canonical homomorphism $A \to k$), R is of degree $\leq d - s$ and $\overline{P} = \overline{Q} \cdot \overline{R}$ (notation of Chapter III, § 4). Show 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 $P = Q_0 R_0$, $\overline{Q} = \overline{Q}_0$, $\overline{R} = \overline{R}_0$ and $Q_0$ contains a monomial $a_0 X_1^s$ where $\phi(a_0) = \phi(a_0)$. (Consider P, Q, R as polynomials in X, with coefficients in the ring B, the completion of $A[X_2, \ldots, X_n]$ with respect to the valuation obtained by extending $v$ by the method of Chapter VI, § 10, no. 1, Proposition 2; then apply Hensel's Lemma; finally use the initial hypothesis on P.)