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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?

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    You're looking for polynomial rings. They're treated in any general abstract algebra text. Here's the wikipedia entry: https://en.wikipedia.org/wiki/Polynomial_ring – Sort of Damocles Jun 03 '23 at 00:17
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    What you're looking for is a text on abstract algebra. In particular, you probably want to look at polynomial rings. At this stage, it's probably a stretch until you take a course on writing mathematical proofs so that you can learn these abstractions. – Sean Roberson Jun 03 '23 at 00:18
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    Oh sorry for the double comment, but the "Not every polynomial being solvable" likely refers to the result that there is not in general a closed form solution to the quintic equation (in contrast with for instance the quadratic equation --> the quadratic formula). This is a result of Galois Theory. – Sort of Damocles Jun 03 '23 at 00:22
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    except for the Galois theory, seems to fit a traditional "Theory of Equations" so here is Dickson, 1922: https://www.gutenberg.org/files/29785/29785-pdf.pdf – Will Jagy Jun 03 '23 at 00:45
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    I advise the book of Prasolov on polynomials. – Jean Marie Jun 03 '23 at 07:10

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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.

B. Goddard
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    (+1) I agree, and I was about to suggest this before I got to your answer. Someone who has just finished Calculus 2 is probably not ready for abstract algebra and Galois theory and such. (moments later) I see now that @Will Jagy also suggested this. I must have skimmed over the comments too fast a few moments ago, as I didn't notice it earlier. Jean Marie's suggestion also seems (possibly) suitable. – Dave L. Renfro Jun 04 '23 at 14:53
  • what are the pre requests for those books – Mathematics enjoyer Dec 25 '23 at 17:30
  • @Mathematicsenjoyer I would think that a typical pre-calc course would be sufficient. – B. Goddard Dec 25 '23 at 18:14