Math

We give a gentle intro to one formalization method – using Lean4 with Mathlib.

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.

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.