Chapter 4 - Quadratic Reciprocity

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.