  Given an integral domain A we consider the set of all integral elements over
A that can occur as an eigenvalue of a symmetric matrix over A. We give a
sufficient criterion for being such an element. In the case where A is the ring
of integers of an algebraic number field this sufficient criterion is also
necessary and we show that the size of matrices grows linearly in the degree of
the element. The latter result settles a questions raised by Bass, Estes and
Guralnick.
