TITLE: Why does the ideal $(a+bi)$ have index $a^2+b^2$ in $\mathbb{Z}[i]$?
QUESTION [10 upvotes]: In the comments on the question Why does this module structure have $352512$ elements?, it is mentioned that the index of the ideal generated by $a+bi$ in $\mathbb{Z}[i]$ has order $a^2+b^2$. 
Is there a nice rigorous explanation of why this is so?

REPLY [2 votes]: If you know a bit of theory, there's a chain of equalities:

The index of the ideal is equal to the number of elements in the quotient ring
The number of elements in the quotient ring is equal to the norm (over $\mathbb{Q})$ of the ideal
The norm of an ideal is equal to the (ideal generated by the) norm of a generator
The norm on $\mathbb{Q}(i)$ over $\mathbb{Q}$ is given by $\mathcal{N}(x + \mathbb{i} y) = x^2 + y^2$.