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- Feb 7, 2012

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$8 = (1 + \sqrt{-7})(1 - \sqrt{-7})$.Let $Z[\sqrt{-7}] ={{a+b\sqrt{-7}}}$ , where a,b are integers. Find 2 irreducible factorisations for 8. I can find one, namely $8=2^3$ but how to find another. More generally, what is the best way of finding irreducible factorisations in general rings?

In a ring of the form $\mathbb{Z}[\sqrt{-p}]$, to factorise an element $a + b\sqrt{-p}$, you need to factorise the