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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.
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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, where
. Then, we have:
Therefore,or
. However, since n is the least positive integer s.t.
, it must be so that
or
, 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)