Congruence relations and congruence modulo m

Question

What are some congruence relations apart from congruence modulo $m$?

Also, is the congruence modulo $m$ called that way because the sets of all the integers $0,1, ..., m-1$ form an isomorphic group under addition mod $ m$ to the group of rotations of an $m$-sided polygon?

Answer 1

An important congruence relation is congruence modulo a polynomial.

For instance, one construction of the complex numbers is $\mathbb R[x] \bmod (x^2+1)$.

More generally, if $p(x)$ is an irreducible polynomial over a field $K$, then $K[x] \bmod f(x)$ is a field containing a root of $f$. (The standard notation is $K[x]/(f(x))$.)