Alladi Ramakrishnan Hall

Siegel norm and character values

Amita Malik

UIUC

Define the length of an algebraic integer $\alpha$ to be the
smallest number of roots of unity which sum up to $\alpha$. In 1969,
Cassels showed that under certain assumptions, an algebraic integer in
an abelian field can be represented as a sum of at most two roots of
unity. Similar results can be obtained for (irreducible) character
values of finite groups. An unpublished theorem of Thompson states
that a character takes the value zero or has length one at more than
one third of the (finite) group elements. We generalize these results
for arbitrary length by establishing a connection between Siegel norm
and the length function. In particular, we obtain a result dual to
that of Burnside. If time permits, I will also talk about an algorithm
to compute cyclotomic integers of length up to a given number.
This is joint work with Florin Stan and Alexandru Zaharescu.