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.

Lecture 17 - MIF; Quadratic Residues
In today's class we finished talking about convolutions, with the highlight being the proof and a few applications of the Mobius Inversion Formula. Afterwards we talked about quadratic congruences, ultimately defining and playing around with so-called quadratic residues.
Lecture 18 - Quadratic Residues; Legendre Symbols
In today's class we discussed some of the basic properties of quadratic residues. We then uncovered the Legendre symbol, exploring its connection to Euler's Criterion and using this to set a game-plan for answering the general question: when is a number a a square modulo a prime p?
Lecture 19 - Gauss' Lemma; Intro to Quadratic Reciprocity
In today's class we began by reviewing the criteria we saw for evaluating Legendre symbols last class period, particularly when the "numerator" was either -1 or 2. Afterwards we set to determine for which primes p we have 2 as a square mod p. This required Gauss' Lemma, which lets us determine whether a is a square based on residues of products ja, where j ranges over the "first half" of residues mod p. After this, we stated the Quadratic Reciprocity Law.
Lecture 20 - Applications of Quadratic Reciprocity
In today's class we covered some applications of quadratic reciprocity, and in particular saw how it could be used to answer whole classes of quadratic congruence questions at once.
Lecture 21 - Proving Quadratic Reciprocity
Today we proved Quadratic Reciprocity. We started by developing a cousin to Gauss' Lemma that's called Eisenstein's Lemma, and then used this lemma — together with a geometric argument — to verify the quadratic reciprocity law.