Christian Roettger: Where are they? ... or: The distribution of normal integral bases in n-space, an exercise in classical representation theory

Let L be a field which is an n-dimensional vector space over Q, the rationals, and let O be the ring of algebraic integers of L (eg L=Q(z), O=Z[z], where z = (1+sqrt{5})/2). Suppose the group G of automorphisms of L has n elements, and that O has a basis over Z of the form {g(a): g in G} for some fixed a in O. In the example above, you could choose a=z. Such a basis is called a normal integral basis, and many people have worked on the question for what fields such a nice basis exists (equivalent to O being a free module over the group ring ZG). Now O can be viewed as a lattice in real n-space. Assuming the existence of a normal integral basis, we can ask how these normal integral bases are distributed in n-space. A rough answer involves only basic principles of classical representation theory. At the end, we will discuss the examples G=S3 (symmetric group of order six) and D4 (symmetry group of the square).

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