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

## Abstract

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.

## Presentation

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