Dr. Keith M. Briggs asked:
I would be grateful for any hints for a solution to this problem:
given an irreducible cubic polynomial p with integer coefficients
and three real roots, find a symmetric (preferably unimodular)
integer matrix with p as its characteristic polynomial, if such exists.
I presume this will involve transformations of the companion matrix.
Alternatively, it would be sufficient to have such a matrix whose
charpoly generates the same field as p.
This appears to be a hard (number theoretic) problem, and GAP does
not seem to help much in its solution. As the mail indicates, there need
not exist a symmetric integer matrix with the given polynomial p as its
characteristic polynomial (an example of such a p is: x^3 - 4x - 1).
I do not know of any method to decide whether a given p as above
is the characteristic polynomial of a symmetric integer matrix.
Best wishes, Gerhard Hiss -- Gerhard Hiss Lehrstuhl D fuer Mathematik, RWTH Aachen Templergraben 64, 52062 Aachen Tel.: (+49) (0) 241 / 80-94543