# A local computation of the Galois group of a polynomial

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 $G$.

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This is sometimes known as "Dedekind's theorem"; that keyword should return some results. –  Qiaochu Yuan Apr 20 '12 at 5:12
@Qiaochu Thanks. I found an article "Tate's Proof of a Theorem of Dedekind". This uses a decomposition group as I suspected. The ground ring of this article is the ring of rational integers. However, by localizing at the prime ideal $pA$, I think the same method can be applied to the above theorem. –  Makoto Kato Apr 20 '12 at 12:58
The proof is given in a generalized form by my answer to [this question][1]. [1]: math.stackexchange.com/questions/111850/a-galois-group-problem –  Makoto Kato Aug 13 '12 at 20:05