Thanks for the improvements and extensions. Merged it.

Btw, since we talked about your style a couple of days ago.

While the article is pretty cool, it is quite hard to understand for somebody without a decent math background (e.g. I would say even a large proportion of beginner CS students will have problems already with the first sentence, where you define a polynomial over a field F).

Also the article is very dry, it's full of mathematical theorems, proofs or proof sketches, by why on earth would anybody ever want to compute the logarithm of a polynomial? Or any of the other operations.
For instance a couple of introductory explanations why polynomials are useful in competitive programming would be very helpful. E.g. by describing that you can often model combinatorial problems by polynomials, by using the number of possibilities as coefficients, and then having nice properties like that you can get the solutions to a product of combinatorial problems by computing the product of combinatorial problems. Like having the polynomial P(x) = 1 + x + x^2 + ... + x^5 for a single dice, and the coefficient of x^100 of P(x)^k is the number of combinations of getting the sum 100 with k dice.

Not sure if it's possible to show the usefulness of the more complex polynomial operations like division, log, ... in a simple way, other than going over the generating functions approaches that you use for problems likehttps://www.codechef.com/MAY20A/problems/RNBWROAD

@adamant-pwn