TITLE: Which positive definite symmetric matrices have solvable characteristic polynomial?
QUESTION [5 upvotes]: I am interested in the structure of the space of $n \times n$ positive definite symmetric matrices with rational entries whose characteristic polynomials are solvable (i.e. the Galois group is solvable).  Is this an algebraic variety, for instance?  I can't find any characterization of such matrices, but I wouldn't necessarily know where to look.

REPLY [6 votes]: Here is a small observation. As a subspace of the space $\mathbb{Q}^n$ of monic polynomials of degree $n$ with rational coefficients, the solvable polynomials are dense (and so in particular are not contained in an algebraic or even semialgebraic subset $\mathbb{Q}^n$). To see this it suffices to observe that any such polynomial is a product of real linear or quadratic polynomials and that we can approximate these by rational linear or quadratic polynomials. The corresponding products are clearly solvable.