Tag: Studies

Let denote the sequence of primes and denote the arithmetic progression with starting term and common difference . Dirichlet’s Theorem on Arithmetic progressions says that if , then contains infinitely many members of . But the Green-Tao Theorem says that for any , there exist such that contains members of . In some sense, these results look like “inverses” of each other from the perspective of sequence containment: at a high level, given two sequences , one result talks about containing members of , and the other talks about containing members of .

Can we make this analogy a bit more precise?

This is a list of definitions and theorems concerning field extensions, that I’m compiling for my learning. Throughout, let be a field.

In typical real analysis classes, the presentation of real numbers that is generally given in schools (starting from rationals) is considered insufficiently rigorous, and replaced by a construction involving Dedekind cuts. However, in this post, we investigate a formalization that derives directly from the presentation given in schools — specifically, using decimals — and study its equivalence to the Dedekind cut formulation.

In ring theory, the characteristic is defined as the min such that , or 0 if no such exists; judging from https://en.wikipedia.org/wiki/Characteristic_(algebra), it seems that no concept of characteristic has been considered for structures more general than rings. Here, we generalize the concept to any monoid .

We attempt to generalize the notion of “normal subgroup,” used in studying kernels and homomorphisms, to a “normal subset” that does not have the requirement of being a subgroup, and we see whether it is possible for a normal subset to exist that is not a normal subgroup.

We list some facts concerning bijections and isomorphisms between algebras for my learning (“algebra” in the universal algebra sense here, not the specific vector space based algebraic structure.)