Tag: Studies

We can think of modular arithmetic as amending operations on Z to “wrap around and stay in Zn.” Can we generalize this idea to arbitrary groups, or even general algebras?

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 .