The following theorem was proved by van der Waerden in his book on algebra. His proof used a factorization of a multivariate polynomial. A paper on the computations of Galois groups said it could be proved by a method of algebraic number theory. I tried to prove it. I thought about using Hensel's lemma but could not follow through. I also thought about using a decomposition group but could not follow through either. How can it be proved?
Theorem Let $A$ be a unique factorization domain. Let $K$ be the field of fractions of $A$. Let $f(X)$ be a monic polynomial in $A[X]$. Supppose $f(X)$ has no multiple root in a splitting field of $K$. Let $G$ be the Galois group of $f(X)$ over $K$. Let $p$ be an irreducible element of $A$. Let $F$ be the field of fractions of $A/pA$. Let $g(X)$ be the reduction of $f(X)$ with respect to $pA$. Suppose $g(X)$ has no multiple root in a splitting field of $F$. Let $H$ be the Galois group of $g(X)$ over $F$. Then $H$ can be embedded in $G$.