If is a unit then there is so that but then thus which means . Thus, the units must have norm equal to . Conversely, if then (which is in ) is inverse of . To find the units we need to solve . This equation is Pellian and got infinitely many solution. I assume all you need to do is find a test to determine if a number is a unit or not, that is how to do it.
A non-zero non-unit element is called irreducible iff is an improper trivial factorization, i.e. one of them, say , is associate to and the other one is a unit. Hence it cannot be reduced further. A non-zero non-unit element is called prime iff . A simple consequence is that prime elements are irreducible, while not necessarily irreducible elements are prime.(b) Show that all four numbers: are irreducible. Are any of these numbers prime?
In a unique factorization domain the irreducible elements are also prime. Thus, if you can find an irreducible non-prime element (which is probably from the above list) then it would mean this is not a unique factorization domain.(c) Deduce that Z(\sqrt10) is not unique factorization domain.