
Christian Roettger: Where are they? ... or:
The distribution of normal integral bases in nspace,
an exercise in classical representation theory
Let L be a field which is an ndimensional 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 nspace. Assuming the existence of a normal integral basis, we can ask how these normal integral bases are distributed in nspace.
A rough answer involves only basic principles of classical representation theory.
At the end, we will discuss the examples G=S_{3} (symmetric group of order six) and D_{4} (symmetry group of the square).
Back
to the seminar schedule

