1. In this exercise we will prove that Z[√-5] is an integral domain but not a unique factorisation domain (UFD). Note that in S we have the following factorisations of the element 6: 6=2.3=(1+√√−5) · (1−√−5). It turns out that these are factorisations into irreducible elements. To see this, prove that the element 1+ √-5 is irreducible, by assuming that 1+√√-5= (a+b√√-5). (c+d√√−5) and taking modulus squared. Explain briefly why a similar argument shows that the other three ele- ments 1-5, 2, and 3 are also irreducible.