Monday, April 16, 2012

You should re-read the stuff you think you know.

I started my PhD in June 2011. While I've only worked on a few problems since then, a pattern is emerging: I always spend a couple months stuck on something that looks obvious and silly (both in fore- and hindsight.) Most recently I was stuck on a problem involving basic arithmetic over finite fields. I felt that it was something I should be able to do, but nonetheless was stuck on it for a few weeks.

Maybe it's that I'm doing the PhD thing wrongly, or maybe it's that I'm not spending enough time on my mathematics (I've got a job doing predictive modeling, too!). In any case, the story of how I got unstuck might interest some people:

I've thought it important to continue doing maths reading independent of my research projects. But while there are several subjects I wish to learn about (and do research in), I have not engaged in much outside study. When I approach a subject, I have two separate reactions: Either I find that the subject is too advanced, and I give up quite early - not wanting to get lost in background material. Or I think the material I have repeats too much of stuff I already know, and I give up - not wanting to waste my time on things I've seen before.

After doing this for a few months, I decided that I should read something that I think I already know - just to get in the habit of things. I started reading the field/Galois theory chapters in the abstract algebra book by Dummit and Foote. Pretty light reading as far as mathematics goes. It's stuff I should know.

Those chapters contain some straightforward proofs of basic statements on algebraic field extensions. Exempli gratia, if $\mu$ is the root of a polynomial $f$, and $\sigma$ is an element of the Galois group, then one can prove that $\sigma(\mu)$ is also a root of the polynomial by observing the action of $\sigma$ on the polynomial $f$. Not particularly new or deep material, but it turns out just the stuff I needed...

Now back to my problems with arithmetic over finite fields: I've spent the last few weeks trying to decompose an operator on polynomials into separate components. My supervisor and I both had our own ideas of writing out matrices for these operators. I spent a few late nights on it, wrote down some messy results, and went to bed hoping I'd have an epiphany on how to clean it up. I had been hoping for one for weeks! But soon after I started re-reading the basics I got it. I was just reading about applying the action of the Galois group to polynomials, and here I was staring at an operator on polynomials. So I abandoned the ``matrix approach'', naively started looking at the action of the Galois group, and a few minutes later I had the result I was looking for. It seemed so obvious in hindsight, but it often seems obvious only after you've understood it.

I should probably keep reading the basics.

1 comment:

  1. I was just thinking how as we we get older and hence wiser in our own view that we always think that we know everything (the perennial know-it-all attitude that has steathily crept up on me) but along the way we have forgotten about the basics which really are the foundation...and everyone knows the story of the house that was built without a foundation...