Integral Domains: Products of Irreducibles

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I'm suppose to find an integral domain where NOT every element (not a unit) is expressible as a finite product of irreducibles.

I'm not sure where to begin, actually. So perhaps someone can give me a tip, and we can start working our way through this. Thanks..
 
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Thank you anyway, all. Since I have other algebra problems that I need help on, I'm going to include this problem in another thread, joining all the problems.
 
closed...see "ALGEBRA PROBLEMS"
 

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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