A Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[ i ]. This is a Euclidean domain which cannot be turned into an ordered ring.

Formally, Gaussian integers are the set

:{a+bi | a,bin mathbb{Z} }.

The norm of a Gaussian integer is the natural number defined as

:N( a  +  bi ) =  a 2 +  b 2.

The norm is multiplicative, i.e.

:N( z · w ) = N( z )·N( w ).

The unit (ring theory)s of Z[ i ] are therefore precisely those elements with norm 1, i.e. the elements

:1, −1, i and − i .

The prime elements of Z[ i ] are also known as Gaussian primes. Some prime numbers (which, by contrast, are sometimes referred to as rational primes ) are not Gaussian primes; for example 2 = (1 +  i )(1 −  i ) and 5 = (2 +  i )(2 −  i ). Those rational primes which are congruent to 3 (modular arithmetic 4) are Gaussian primes; those which are congruent to 1 (mod 4) are not. This is because primes of the form 4 k  + 1 can always be written as the sum of two squares (Fermat s theorem on sums of two squares), so we have

: p  =  a 2 +  b 2 = ( a  +  bi )( a  −  bi ).

If the norm of a Gaussian integer z is a prime number, then z must be a Gaussian prime, since every non-trivial factorization of z would yield a non-trivial factorization of the norm. So for example 2 + 3 i is a Gaussian prime since its norm is 4 + 9 = 13. This implies that since there are infinitely many ordinary primes then there must be infinitely many Gaussian primes.

The ring of Gaussian integers is the integral closure of Z in the field (mathematics) of Gaussian rational Q( i ) consisting of the complex numbers whose real and imaginary part are both rational number.

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