# Section 5.1: Analysis

## Congruence in $F[x]$$F[x]$ and Congruence Classes

This section takes all of the ideas of congruence and congruence classes from section 2.1 and demonstrates how it is valid in the field $F[x]$. The only visible adjustment is the definition of a residue class which essentially is the set of polynomials that leave a specific polynomial remainder when divided by another polynomial.

I wonder, however, how this can be useful, that is how is a residue class useful. It seems to be the equivalent of the remainder in $\mathbb{Z}$ after applying the division algorithm. If it is different how is it different? If it isn’t then why is it given a name other than ‘remainder polynomial’ or something along those lines.