Some classical results in number theory

The law of quadratic reciprocity

Nowadays, the most commonly seen proof of the Law of Quadratic Reciprocity is Eisenstein's reformulation of Gauss's third proof. We present some other classical proofs.

  1. Gauss's fifth proof. This is a completely elementary proof, relying, as Gauss's third proof did, on Gauss's lemma. Available in pdf format.

  2. A proof by Jacobi, with a simplification by Legendre. This proof begins with a consideration of Gauss sums. (It only uses the value of the square of a Gauss sum, not the determination of the sign of the square root, which is much deeper.) Available in pdf format.

The Jacobi symbol

  1. An exposition of the basic properties of the Jacobi symbol, together with a method of calculating Jacobi symbols, due to Eisenstein. This of course gives a method for computing Legendre symbols as well. Available in pdf format.

The irreducibility of the cyclotomic polynomials

We present several proofs of the irreducibility of the cyclotomic polynomials.

  1. A compilation of a number of classical proofs, due to Gauss, Kronecker, Schönemann/Eisenstein, Dedekind, Landau, and Schur. Available in pdf format.

  2. A (to the author's knowledge) new proof motivated by Gauss's original proof. Available in pdf format.