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).

We give definitions for span, basis and dimension, and then prove that all vector spaces have bases, and that their dimension is well-defined. Then we use that to define the degree of a field extension and prove the tower law for field extensions. After that, we define basic properties of polynomials.

In this entry we cover more basic results about ordinal arithmetic. We also prove the so-called “Fundamental Theorem of Cardinal Arithmetic”, and then we finish with a short discussion about cofinality of cardinals.

Here we dig into ordinal and cardinal arithmetic a little bit. We also establish the famous result (originally due to Cantor) that |S| < |P(S)|, i.e. that the power set of any set is larger than the set.