## Happenings May 8

Well, last Saturday morning I woke up determined to do mathematics, and knowing exactly what I was going to do. It was a very productive day, but at the end of it I was completely worn out. I didn’t stop to write about what was happening, and I didn’t let the kid do any mathematics of his own.

Not for a while, anyway. He has moved on to finite fields. There are two interesting facts that served as a starting point.

First, consider the set $Z_n$ of integers mod n. The first interesting fact is that $Z_n$ is a field if and only if n is prime. For example, $Z_3 = \{0,1,2\}$ with multiplication and addition mod 3 is a field. $Z_4 = \{0,1,2,3\}$ is not – because 2*2 = 0, so 2 does not have a multiplicative inverse.

So, “we know” that there exists a finite field with p elements if p is prime. The second interesting fact is that there also exists a finite field with $p^n$ elements if p is prime, for any positive integer n.

We could combine these two facts and a little more and say that there is a unique finite field of order $p^n$ for all primes p and all positive integers n.)

The challenge is that the case n = 1 is very easy, but the cases n > 1 are not.

In fact, until yesterday I had never held a finite field with n > 1 in my hands. Well, now I know a representation for the finite field of order $4 = 2^2\$. I’m looking forward to holding more finite fields in my grubby little hands.

In the meantime, I have implemented the dyadic expansion of a scaling function and of the corresponding mother wavelet, for several cases. You should be seeing them in the next couple of posts.

(OK, the dyadic expansion of the D4 scaling function is out there; the corresponding calculation for the D4 mother wavelet should be next; and more examples should follow quickly.)

It took longer than I expected to put out the post – it usually does. And it took some editing afterwards. I had forgotten that sometimes the LaTeX actually renders, but incorrectly. In particular, < stuff > gets treated as (silly) HTML and vanishes. (And, yes, I had to code those angle brackets – I can’t just type them. Grrr.) I prefer it when I get the yellow banner that says it couldn’t render at all.

One of the reasons for making sure that I get this post out tonight is that I don’t want to have to stop to do this tomorrow morning.