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.

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.

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
 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.

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

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