Ring Theory: Show Phi(a)= a^p is Isomorphism

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Homework Statement


Given a commutative ring R with a prime characteristic p, show that the mapping phi:R-->R defined by phi(a)= a^p is a isomorphism

Homework Equations


Fermat's little theorem(I think)

The Attempt at a Solution



I'm pretty sure Fermat's theorem must have something to do with this. Or I could be completely wrong, it just popped into my head very quickly and I haven't really been able to find out if it applies in this case (some sort of generalization, at least), how to show it applies, or come up with any other idea.

Also, sorry for the formatting, I don't know LaTeX
 
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You will want to consider \phi(a+b) \mod p.
 
roight
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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