Abstract Algebra/Polynomial Rings

Although there is a theory of non-commutative polynomial rings, it presents some difficulties and will not be treated on this page. Thus, we will work only with commutative rings for their polynomial rings.

The degree of a polynomial $a_{0}+a_{1}X+...+a_{n}X^{n}$ is defined to be $n$. If $R$ is a field, and $f$ and $g$ are polynomials of $R[X]$, then we can divide $f$ by $g$ to get $f=gq+r$. However, we can also do this for any arbitrary ring if the leading coefficient of $g$ is 1.