Gaussian Integer - As A Principal Ideal Domain

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.