Gaussian integer
A Gaussian integer iscomplex number whose realimaginary partboth integers. The Gaussian integers,ordinary additionmultiplicationcomplex numbers, form an integral domain, usually written as Z[i]. This isEuclidean domain which cannot be turned into an ordered ring.The norm ofGaussian integer isnatural number defined as N(a + bi) = a2 + b2. The normmultiplicative, i.e. N(zw) = N(z)N(w). The unitsZ[i]therefore precisely those elementsnorm 1, i.e.elements 1, -1, i-i.
The prime elementsZ[i]also known as Gaussian primes. Some prime numbersnot Gaussian primes;example 2=(1+i)(1-i)5=(2+i)(2-i). Those prime numbers whichcongruent3 mod 4Gaussian primes; those whichcongruent1 mod 4not. Thisbecause primes ofform 4k+1 can always be written assumtwo squares, so we have p = a2 + b2 = (a + bi)(a - bi). Ifnorm ofGaussian integer z isprime number, then z must beGaussian prime, since every non-trivial factorizationz would yieldnon-trivial factorization ofnorm. Soexample 2 + 3i isGaussian prime since its norm4 + 9 = 13.
The ringGaussian integers isintegral closureZ infieldGaussian rationals Q(i) consisting ofcomplex numbers whose realimaginary partboth rational.
