Hopf Orders

Abstract

To study the integral Galois module structure of wildly ramified extensions of number fields, we will need to work with orders in the group algebra that are larger than the group ring. We shall consider only those orders of a special type, namely Hopf orders. We will only be concerned with orders in either a group algebra or the algebra of maps from a group to the ground field, and for ease of exposition we will only define Hopf orders in the context of these particular Hopf algebras. After giving various general algebraic results about such Hopf algebras and Hopf orders, we will show how to classify all Hopf orders in the group algebra of a group of prime order, and will briefly discuss how this technique can be generalised to a special class of Hopf orders in the group algebra of an elementary abelian group. In the final section of this chapter, we will indicate how commutative Hopf algebras in general have a geometric interpretation in terms of affine group schemes, and will show how examples of Hopf orders can be constructed using formal groups.