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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 $\displaystyle n=st$, where $\displaystyle 1\leq s,t\leq n$. Then, we have:
$\displaystyle 0=n\cdot 1=(st)\cdot 1=(s\cdot 1)(t\cdot 1)(*)$
Therefore, $\displaystyle s\cdot 1=0$ or $\displaystyle t\cdot 1=0$. However, since n is the least positive integer s.t. $\displaystyle n\cdot=0$, it must be so that $\displaystyle s=n$ or $\displaystyle t=n$, 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)
