books on polynomial algebra with much detail than pre-calc books

Question

I just finished a course in Calculus 2, and I was surprised that there are many theorems that consider pre-calculus theorems like "theorems of polynomials" such as the Fundamental Theorem of Algebra, Descartes' Rule of Signs, the Factor Theorem,..... . but We didn't study these theorems in high school or college. However, after finishing Calculus 2, pre-calculus books seem too easy for me. Reading one seems like a waste of time. Although some of the the theorems are very basic that I can learn them just from youtube, like the Factor Theorem, I am pretty sure that there is much in the field of basic algebra that covers only polynomials and is not elementary. It considers pretty advanced topics like not every polynomial being solvable. So my question is: Is there a book that covers "theorems of polynomials" with much depth, detail, and rigor and what course cover these topics?

Answer 1

I think the topic you're looking for is "theory of equations." It's sort of old-fashioned, and has been absorbed by algebra, but there are still texts and websites that treat the subject. Here's the wiki page:

https://en.wikipedia.org/wiki/Theory_of_equations

and here's a link to a book that's 120 years old, but probably has the sort of stuff you're looking for:

https://www.amazon.com/Introduction-Algebraic-Equations-Leonard-Dickson/dp/B00AOX1S7C/ref=sr_1_6

As others are suggesting, you could take the algebraic route (and you should, eventually) but "theory of equations" seems more direct to your question.