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 introduce localization of a ring and the ring of fractions of a ring. We then prove that the ring of polynomials over a field is a Euclidean Domain, and then finish by proving Gauss’ Lemma that a polynomial is irreducible over a ring exactly when it is irreducible over the ring’s field of fractions.
We prove more results for Ordinal and Cardinal arithmetic. Including showing that we can apply division with remainders to ordinals. We use that to prove Cantor’s Normal Form Theorem. Finally, we give a proof of Konig’s Theorem for Cardinals.
In this entry, we define the sum and product of two ring Ideals. We then prove the Chinese Remainder Theorem for Ideals. Along the way, we study a tiny bit of Category theory, enough to define free objects and work a little bit with free rings.
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.
