  We introduce a notion of "Galois closure" for extensions of rings. We show
that the notion agrees with the usual notion of Galois closure in the case of
an S_n degree n extension of fields. Moreover, we prove a number of properties
of this construction; for example, we show that it is functorial and respects
base change. We also investigate the behavior of this Galois closure
construction for various natural classes of ring extensions.
