**As A Principal Ideal Domain**

The Gaussian integers form a principal ideal domain with units 1, −1, *i*, and −*i*. If *x* is a Gaussian integer, the four numbers *x*, *ix*, −*x*, and −*ix* are called the associates of *x*. As for every principal ideal domain, the Gaussian integers form also a unique factorization domain.

The prime elements of **Z** are also known as **Gaussian primes**. An associate of a Gaussian prime is also a Gaussian prime. The Gaussian primes are symmetric about the real and imaginary axes. The **positive integer** Gaussian primes are the prime numbers congruent to 3 modulo 4, (sequence A002145 in OEIS). One should not refer to only these numbers as "the Gaussian primes", which term refers to **all** the Gaussian primes, many of which do not lie in **Z**.

A Gaussian integer is a Gaussian prime if and only if either:

- one of
*a*,*b*is zero and the other is a prime number of the form (with*n*a nonnegative integer) or its negative, or - both are nonzero and is a prime number (which will
*not*be of the form ).

The following elaborates on these conditions.

2 is a special case (in the language of algebraic number theory, 2 is the only ramified prime in **Z**).

The integer 2 factors as as a Gaussian integer, the second factorisation (in which *i* is a unit) showing that 2 is divisible by the square of a Gaussian prime; it is the unique prime number with this property.

The necessary conditions can be stated as following: if a Gaussian integer is a Gaussian prime, then either its norm is a prime number, or its norm is a square of a prime number. This is because for any Gaussian integer, notice

- .

Here means “divides”; that is, if is a divisor of .

Now is an integer, and so can be factored as a product of prime numbers, by the fundamental theorem of arithmetic. By definition of prime element, if is a Gaussian prime, then it divides (in **Z**) some . Also, divides

- , so in
**Z**.

This gives only two options: either the norm of is a prime number, or the square of a prime number.

If in fact for some prime number, then both and divide . Neither can be a unit, and so

- and

where is a unit. This is to say that either or, where .

However, not every prime number is a Gaussian prime. 2 is not because . Neither are prime numbers of the form because Fermat's theorem on sums of two squares assures us they can be written for integers and, and . The only type of prime numbers remaining are of the form .

Prime numbers of the form are also Gaussian primes. For suppose for, and it can be factored . Then . If the factorization is non-trivial, then . But no sum of squares of integers can be written . So the factorization must have been trivial and is a Gaussian prime.

If is a Gaussian integer whose norm is a prime number, then is a Gaussian prime, because the norm is multiplicative.

Read more about this topic: Gaussian Integer

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