Separable polynomials over an
arbitrary commutative ring are studied. Given any separable polynomial
p(X) over the commutative ring R one can find a “splitting ring” for p(X)
which is a finitely generated normal separable extension of R generated by
roots of p(X). A polynomial closure Λ of R generated by roots of separable
polynomials is constructed. Any separable polynomial over Λ factors into linear
factors in Λ. A Galois theory for such extensions is discussed. Applications to
separable extensions of von Neumann regular rings and the Brauer group are
given.
