Commutative Diagrams Explained

Commutative Diagrams Explained

Have you ever come across the words "commutative diagram" before? Perhaps you've read or heard someone utter a sentence that went something like

"For every [bla bla] there exists
a [yadda yadda] such that
the following diagram commutes."

and perhaps it left you wondering what it all meant.

Read More

Resources for Intro-Level Graduate Courses

Resources for Intro-Level Graduate Courses

In recent months, several of you have asked me to recommend resources for various subjects in mathematics. Well, folks, here it is! I've finally rounded up a collection of books, PDFs, videos, and websites that I found helpful while studying for my intro-level graduate courses.

Read More

What is Galois Theory Anyway?

What is Galois Theory Anyway?

Perhaps you've heard of Évariste Galois? (Pronounced "GAL-wah.") You know, the French mathematician who died tragically in 1832 in a duel at the tender age of 20? (Supposedly over a girl! C'est romantique, n'est-ce pas?)  Well, today we're taking a bird's-eye view of his most well-known contribution to mathematics: the appropriately named Galois theory. The goal of this post is twofold... 

Read More

Rational Canonical Form: Example #2 (with Galois Theory)

Rational Canonical Form: Example #2 (with Galois Theory)

Last week we saw an example of how to use the rational canonical form (RCF) to classify matrices of a given order in GL_2(Q). Today we have a similar example (taken from CUNY's spring 2015 qualifying exam) where now our matrices have entires in the finite field F_13. The fact that our field is F_13 instead of Q actually makes little difference in how to approach the solution, but I think this problem is particularly nice because part of it calls on some Galois Theory. 

Read More