## Introduction to Ideal Class Groups

Algebraic number theory is a really interesting subject, but unlike some other subjects, it’s not 100% clear what objects people study. This post provides an introduction to the class group of a finite dimensional field extension of $\mathbb{Q}$, an object often … Continue reading

Posted in Math, Number Theory, Uncategorized | Tagged , | 1 Comment

## A Pretty Lemma About Prime Ideals and Products of Ideals

I was trying to prove a theorem in algebraic geometry which basically held if and only if this lemma held. Here’s the lemma: Lemma: Given any ring $A$, a prime ideal $\mathfrak{p} \subset A$, and a finite collection of ideals … Continue reading

Posted in Algebra, Algebraic Geometry, Math | Tagged , , | 3 Comments

## The “Idea” of a Scheme

The mathematical concept of a “scheme” seems to pop up everywhere, but it’s hard to get a good grasp on what a scheme actually is. Any time you might ask someone what a scheme is in passing, there never seems … Continue reading

Posted in Algebraic Geometry, Math | Tagged , | 1 Comment

## Ordered Fields and When You Can’t Order Them

The real numbers have an ordering on them–given two numbers and , we can tell whether  or . So as math people, we like to generalize this to other sets–when can we say that a general set is ordered? In this post, … Continue reading