Notes: Number Theory and Modern Algebra I

I took Modern Algebra I with Professor Matthias Beck in Fall 2015. These comprehensive notes were compiled using lecture notes and the textbooks,

Please feel free to download and print these notes for your convenience.

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

Topics Covered:

  • Integers & the Euclidean algorithm
  • Complex numbers, roots of unity & Cardano’s formula
  • Modular arithmetic & commutative rings
  • Polynomials, power series & integral domains
  • Permutations & groups

Featured Image: Dodecahedron-Icosahedron Duality
Credit: Images from Algebra: Abstract and Concrete by Frederick M. Goodman

Website Infrastructure

The notes page is set up, next I aim to finish up organizing the precalculus resources and general non-math resources.

Now, the site contains links all my notes starting from Fall 2015, which send visitors directly to my google drive, but I wonder if that is secure/wise.

Maybe there is a better way to both host the files and allow for instant updates as I compile my LaTeX files. Right now, this blog links to individual files within my google drive which is synced to a directory on my laptop. When I compile on my computer, I work directly in that directory so that the file linked in google drive is automatically synced. I want the same functionality and convenience but I’m not sure if linking to the drive is secure, since it’s a personal account.

Time to do some non-math research~

Update: I hear that wordpress has great support, so I made a post at the forums. I wonder if there’ll be an easy solution?

Remark about Self-Care

Let \mathscr{H} be the set of all humans.

Definition: If an element x “cares for” a set S, it “cares for” every element in the set.

Remark: If you are a human, and you “care for” humanity, then you must “care for” yourself too.

Proof of Remark: Let x denote you. x \in \mathscr{H}; x “care for” \mathscr{H} \implies x “care for” x.

Why a Blog?

So, why start a blog when there is approximately… 1.5 \times 10^{6} posts created per day? (Checked @ 8 AM 1/8/17)

For me, this website is a bit more than a collection of math notes and lists of resources. I am aiming to chronicle the challenges of being a student and a teacher at the same time. On the other hand, I want to also document the most fun parts of being in an intensive graduate program!

So this blog will be part survival guide, part chicken-soup-for-the-soul, and part scrapbook.

Oh. By the way, I survived the first semester of graduate school!