UC Santa Cruz

In this thesis we define the notion of a Galois extension of commutative rings, and present the analogue of the fundamental theorem of Galois theory in this setting. Following the work of Chase, Harrison, and Rosenberg, we show how the classical definition of a Galois extension of a field arises as a special case of this generalization. Furthermore, we generalize the notion of a Galois extension of commutative rings by replacing the Galois group with a Hopf algebra, leading to the notion of a Hopf Galois extension. We present the fundamental theorem in this context and show how the definition of a Galois extension of a commutative ring arises as a special case of this generalization.