The Routh-Hurwitz criterion and method are usually taught in a cookbook format. Essentially, you follow a recipe for placing the coefficients into a table and perform "figure 8" multiplication and subtraction (like the tabular method for computing cross products in vector math). This recipe produces easy visual cues for determining stability, instability, and the presence of special cases like a factor polynomial known as the auxiliary polynomial. While it works great, I'm really curious as to how/why it works from a pure math perspective.
My question is: what is the name or source of the underlying theory of polynomial arithmetic that one can consult in order to understand this method at a deeper level? Internal questions would be: How are the figure-8 expressions justified? Are they an application of some more complicated theory of factoring and analyzing polynomials? What is the significance of alternating even and odd-powered terms?
Providing an online pointer to the original Routh-Hurwitz criterion paper or its citations would be great, but even just naming the theory or literary source that pertains to the analysis of polynomials would be very helpful.
Please don't provide links to another "applied math" exposition of Routh-Hurwitz nor a recent re-proof of it using esoteric pure math techniques. What I'm looking for is a basic pointer to the original theory that supports it.