Chapter 4 is all about determining which numbers modulo p (where p is an odd prime) are actually squares of other numbers modulo p. This culminates in Gauss's amazing Law of Quadratic Reciprocity.

In today's class we began by talking about quadratic residues once again. We noticed that we could list the quadratic residues by squaring the first $\frac{p-1}{2}$ residues mod p, and that there were an equal number of quadratic residues and nonresidues. We then introduced the Legendre symbol $\left(\frac{a}{p}\right)$ as the "square indicator function modulo p". Finally we discussed Euler's Criterion for evaluating the Legendre symbol $\left(\frac{a}{p}\right)$ and a few of its consequences.