# integral domains, fermats

• Apr 12th 2009, 05:38 PM
lttlbbygurl
integral domains, fermats
a) Show that the characteristic of an integral domain D must be either 0 or prime.
b) Show that 1 and p-1 are the only elements of the field Zp that are their own multiplicative inverse.
c) Use Fermat's theorem to show that for any positive integer n, the integer n^(37) - n is divisible by 383838.

(:
• Apr 12th 2009, 07:24 PM
Dark Sun
Hello littlebbygurl, I have the proof of a) for you:

If 1 has infinite order under addition, then the characteristic of the integral domain is 0.

Suppose that 1 has order n, and $n=st$, where $1\leq s,t\leq n$. Then, we have:

$0=n\cdot 1=(st)\cdot 1=(s\cdot 1)(t\cdot 1)(*)$

Therefore, $s\cdot 1=0$ or $t\cdot 1=0$. However, since n is the least positive integer s.t. $n\cdot=0$, it must be so that $s=n$ or $t=n$, proving that n is prime.

If (*) is not trivial to you, you may want to prove it as well. The same is true of line one of the proof.

Hope this helps, happy mathing! (Cool)