Abstract Algebra/Polynomial Rings

From Wikibooks, open books for an open world
Jump to: navigation, search

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_1X+...+a_nX^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.