I saw Perron's criterion( which is lovely), I was wondering if it can be changed in the following way (so I have the idea for what the proof will be, but not how to find a polynomial to satisfy the needed properties):
Instead of showing all but 1 root lie outside the unit circle, we show that all but 2 lie outside the circle, and we also show those 2 aren't real, it follows that if the polynomial was reducible those 2 would be in the same factor and so it isn't reducible.
Could someone help me build such a general polynomial?
EDIT:
So I want a polynomial $ a_n*x^n +a_{n-1}*x^{n-1} ...+a_0$ so that $|a_{n-2}| > |a_n|+|a_{n-1}|+|a_{n-3}|...+|a_0|$ and also that it won't have real roots (specifically that it won't have 2 real roots outside of the unit circle), and none of the roots will be on the unit circle, what's the most general one I can find?