Chapter 4: Quadratic Residues

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: Legendre Symbols and Euler's Criterion

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 residues mod p, and that there were an equal number of quadratic residues and nonresidues. We then introduced the Legendre symbol as the "square indicator function modulo p". Finally we discussed Euler's Criterion for evaluating the Legendre symbol and a few of its consequences.

Lecture 19: Gauss' Lemma; 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 proving our rule for determining whether 2 is 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 and applied the Quadratic Reciprocity Law.

Lecture 20: Applications of Quadratic Reciprocity; Eisenstein's Lemma

Today we talked more about quadratic reciprocity. We started by giving some more applications of this powerful theorem. We then concluded class by giving a proof of Eisenstein's Lemma, a sort of cousin of Gauss' Lemma.

Lecture 21: A Proof of Quadratic Reciprocity; Order Calculations

Today we finished our discussion of quadratic reciprocity by providing a proof of the theorem which is based on Eisenstein's Lemma. Afterwards we began talking about the notion of order for a given integer a modulo m, as well as what it meant for a to be a primitive root modulo m. We calculated the order of a few integers, and we began talking about one of the basic divisibility properties of order.