We prove that Z is universal in the category of rings with identity. Using that, we define the characteristic of a field, and prove that it is well-defined. We then discuss ring ideals a bit more before finishing by showing that all fields with exactly p elements (where p is prime) are isomorphic.
We define the Fibonacci Sequence, develop a formula for the entries. We then use that to establish bounds on the growth of the sequence. We use that to prove a bound on the number of division operations required to compute the Euclidean Algorithm. Finally, we finish by continuing our discussion of the RSA algorithm and introducing the Golden Mean.
In this entry we begin by proving the correctness of the Euclidean Algorithm, then we discuss factorization in rings. We give an example of factoring into irreducibles in two distinct ways. We then define Unique Factorization Domain, Principal Ideal Domain and Euclidean Domain. We then prove that an ED is a PID and a PID is a UFD. We finish by explaining the RSA encryption algorithm.
We continue our investigation into rings and fields. We finish by explaining the Euclidean Algorithm. We also give a python implementation which, for any two positive integers, a and b, returns gcd(a,b) and the pair of integers, s and t, such that as + bt = gcd(a,b).
