## 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

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## 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

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## 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

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## On “Imposter Syndrome”

Here’s how it happens: You’re in graduate school and were one of the best people in your major from your school. Honestly, that’s how you got into graduate school in the first place. You go in the first few weeks, you … Continue reading