Theory of numbers: Quadratic reciprocity

This lecture is part of an online undergraduate course on the theory of numbers.

We state and law of quadratic reciprocity for Legendre symbols, and prove it using Gauss sums. As applications we show how to use it to calculate Legendre symbols and to test Fermat numbers to see if they are primes.

Update: As pointed out in a comment, the estimate of the number of proofs of the quadratic reciprocity law in the lecture is out of date. The book "The Quadratic Reciprocity Law" by Oswald Baumgart lists 314 proofs as of 2014.

[schrodingerbracat]

Nice to see you. I was one of your supervision students in Cambridge 1989 for Galois Theory, that was before you won the Fields Medal. One of our sessions went on for hours even though it was supposed to be only one hour. You have always been very passionate about maths and teaching maths.

[sajjadakbar]

the number of proofs (counting slightly different proofs to be entirely different proofs) is 314 dated by 2014, this is given in the 2015 edition of the same book.

[dmitripanov]

It seems to me that around 8.30 when you prove tau=1+2alpha you use the equality 1+eps+eps^2+...+eps^p-1=0 without mentioning it?

[algebraist]

Will you be making a video series for class field theory (with Galois cohomology)?

[datsmydab-minecraft-and-mo]

What books would you recommend for number theory