Why taking difficult courses is worth the effort

So I have to admit, my grade in algebraic number theory is lacking in goodness right now. I’m barely passing the course, and that’s only after scoring 8.5 / 10 on the previous two homeworks. To note, 6.0 / 10 is passing on the Dutch scale.

Not to mention the fact that I constantly work on this course. I’ve put in at leave 20 hours per week into this course. It’s a good thing that my other course was easy, it allowed me to travel around the country.

Yet, I’m really glad I took this course.

I’m caught up to the point of understanding nearly everything in the course. Well, I still suck at the questions and can only do the “easy” ones, but I’m doing those better than I was doing the previous easy exercises. This is namely due to the fact that I should have had at least one semester of graduate commutative algebra before taking this course. A number theory course would have been beneficial as well.

But anyhow, it’s worth taking the course because it’s something with so many open problems.

See, we’ve gone pretty quick in this course. So quick, that the instructor is having to slow down even for the graduate students, not just my benefit. This is great though, because we’re at the point of reaching open questions.

An open question is a question that has been posed by some researcher in the past. There is great open questions, posed by some brilliant men hundreds of years ago. The last great open question was solved when Wiles proved Fermat’s Last Theorem. This proof actually relied heavily on algebraic number theory, and if I had courses in elliptic curves and complex functions, I could probably be able to mostly understand it.

But there’s other questions that don’t don’t rely on such elaborate techniques to understand. Some of these are what we are approaching in my algebraic number theory course. While I’m not at the level of being able to solve these problems, I can understand their beauty. That’s right, the best questions look beautiful, and have beautiful solutions.

So that’s why it’s worth it to push yourself and take a course that is tougher. You stretch your brain.