2.2

1. The most difficult thing I can foresee is proving that Z modulo n is only a field if n is a prime number. Actually, that won't be that bad, since the greatest common divisor of the prime number and all its congruence classes is 1 (except for [0]). I'm taking Theory of Analysis (315) and I have to prove, with fixed b > 0, that r = m/n = p/q => (b^m)^(1/n) = (b^p)^(1/q). Any suggestions?

2. I was talking to a BYU mathematics Master's student and he told me he could find no application for the infinity of infinities proof (that there are uncountable sets of uncountable sets). It got me thinking about all of the development of prime numbers and number theory in a time when it was unreasonable to use them for, say, cryptography. I can't see a use for finite fields yet, but I'm sure when I realize what I can do with them, I'll wonder what I ever did without them.