Editing Gaussian integer (section)

==Gaussian primes==
As the Gaussian integers form a [[principal ideal domain]], they also form a [[unique factorization domain]]. This implies that a Gaussian integer is [[irreducible element|irreducible]] (that is, it is not the product of two [[unit (ring theory)|non-unit]]s) if and only if it is [[prime element|prime]] (that is, it generates a [[prime ideal]]).

The [[prime element]]s of {{math|'''Z'''[''i'']}} are also known as '''Gaussian primes'''. An associate of a Gaussian prime is also a Gaussian prime. The conjugate of a Gaussian prime is also a Gaussian prime (this implies that Gaussian primes are symmetric about the real and imaginary axes).

A positive integer is a Gaussian prime if and only if it is a [[prime number]] that is [[Congruence class|congruent to]] 3 [[modulo operator|modulo]] 4 (that is, it may be written {{math|4''n'' + 3}}, with {{math|''n''}} a nonnegative integer) {{OEIS|A002145}}. The other prime numbers are not Gaussian primes, but each is the product of two conjugate Gaussian primes.

A Gaussian integer {{math|''a'' + ''bi''}} is a Gaussian prime if and only if either:
*one of {{math|''a'', ''b''}} is zero and the [[absolute value]] of the other is a prime number of the form {{math|4''n'' + 3}} (with {{mvar|n}} a nonnegative integer), or
*both are nonzero and {{math|''a''<sup>2</sup> + ''b''<sup>2</sup>}} is a prime number (which will ''never'' be of the form {{math|4''n'' + 3}}).

In other words, a Gaussian integer {{math|''m''}} is a Gaussian prime if and only if either its norm is a prime number, or {{math|''m''}} is the product of a unit ({{math|±1, ±''i''}}) and a prime number of the form {{math|4''n'' + 3}}.

It follows that there are three cases for the factorization of a prime natural number {{math|''p''}} in the Gaussian integers:
*If {{math|''p''}} is congruent to 3 modulo 4, then it is a Gaussian prime; in the language of [[algebraic number theory]], {{math|''p''}} is said to be [[inert prime|inert]] in the Gaussian integers.
*If {{math|''p''}} is congruent to 1 modulo 4, then it is the product of a Gaussian prime by its conjugate, both of which are non-associated Gaussian primes (neither is the product of the other by a unit); {{math|''p''}} is said to be a [[decomposed prime]] in the Gaussian integers. For example, {{math|1=5 = (2 + ''i'')(2 − ''i'')}} and {{math|1=13 = (3 + 2''i'')(3 − 2''i'')}}.
*If {{math|1=''p'' = 2}}, we have {{math|2 {{=}} (1 + ''i'')(1 − ''i'') {{=}} ''i''(1 − ''i'')<sup>2</sup>}}; that is, 2 is the product of the square of a Gaussian prime by a unit; it is the unique [[ramification (mathematics)#In algebraic number theory|ramified prime]] in the Gaussian integers.