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 $\frac{p-1}{2}$ 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.
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.
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