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.

page revision: 11, last edited: 20 Oct 2008 19:29