#### 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 An unpublished theorem of Thompson states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.

Done