Ben asked in class 11/13 whether it's a coincidence that the ring Z[i]/<2+i> has 5 elements and the square of the modulus of 2 + i is 5 (example 4 on page 142 of our textbook). No, it is not a coincidence!

THEOREM: If z = a + bi is a non-zero element of the ring of Gaussian integers Z[i], then the quotient ring Z[i]/<z> has n elements, where n is the modulus of z, n = a² + b².

This theorem is a corollary of Pick's Theorem.