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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.

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    $\begingroup$ What do you know about polynomials whose Galois group is solvable? $\endgroup$
    – Igor Rivin
    Commented Nov 18, 2013 at 15:21
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    $\begingroup$ Surely the space of positive definite symmetric matrices already fails to be an algebraic variety. It's a semialgebraic set, though. $\endgroup$
    – Qiaochu Yuan
    Commented Nov 18, 2013 at 18:32

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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.

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  • $\begingroup$ That's a good observation; I had hoped to establish a fixed point principle on the space of "solvable matrices", but your remark suggests that this is hopeless. $\endgroup$
    – Paul Siegel
    Commented Nov 19, 2013 at 23:29

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