The
reason for not allowing
1 as
prime is to keep the
fundamental theorem of arithmetic. If you could say 24=2*2*2*3=1*1*1*1*2*2*2*3 were both decompositions of
24 into
prime factors, the decomposition would not be
unique.
Pakaran is incorrect regarding "prime" complex numbers. The integer ring of the algebraic numbers is the ring of gaussian integers -- numbers of the form a+bi with a,b both integers. In that ring, as Noether's excellent gaussian integers writeup shows, an integer p that is prime as an integer is prime iff it is not of the form 4k+1. When p=4k+1, Noether shows it is possible to write p = a2+b2 = (a+bi)(a-bi),
so p is not prime. But since the norm squared (absolute value) of a±bi is p, and since the norm squared of any gaussian integer is integer and multiplicative, it follows that a±bi
itself is a prime gaussian integer.
Thus we can classify all prime gaussian integers.