Properties of Polynomial Rings Over Various Rings: Zp, Z, R and C.

Levi Puga and Dr. Ivona Grzegorczyk


We will analyze and study polynomial rings over Zp, Z, R and C(with one variable). More specifically we will analyze criteria for divisibility, that is, when a polynomial is divisible by another(is a factor) and when a polynomial is irreducible(has “no” roots/factors). Thanks to certain theorems we can see that polynomials in certain rings will always have roots/factors and therefore that polynomial will be divisible by that said factor. For instance, due to the Fundamental Theorem of Algebra we can say that a polynomial of degree n will have n roots in the Complex Numbers(C). How about that same polynomial in R? In Z? In Zp? It is also important to see the relationship between being divisible(remainder=0) and having a factor. If a polynomial has a factor then it is divisible by that factor, but not all polynomials will have factors(polynomial is irreducible) We will analyze these questions and more.


  1. Congratulations Levi! I enjoyed the animated examples – great choice! Can you speak about why this topic is so important in applications?

    • Hi Prof Flores! Of course I can. Many things in he real world can be modeled by polynomial equations and finding out when those polynomials equal zero can give us some valuable information. Something fun would be modeling the path/journey of a roller coaster. Let’s say an amusement park owner wants to build a new coaster that goes underwater and pops back up onto the surface and the path of the coaster is modeled by c(x)=x^5-4x^3+2x^2-3x-2. If input is time and output is height we will be able to find out when the coaster will go underwater and pop back onto the surface by reducing/factoring c(x). Now if c(x) does in-fact travel in and out of the surface/water then it must cross the x-axis(surface level)at some point. By using the RRT and Synthetic Division we can actually see that c(x) is reducible over Q(Z aswell) and c(x)=(x+2)(x+1)(x-1)^3. Now we know when the roller coaster will pop in and out of the surface. In summary: Finding out if a polynomial is reducible/factorable/divisible can help us discover valuable information especially if that polynomial is modeling something.

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