Editing Gaussian integer (section)

==Basic definitions==
The Gaussian integers are the set<ref name="Fraleigh 1976 286"/>
:<math>\mathbf{Z}[i]=\{a+bi \mid a,b\in \mathbf{Z} \}, \qquad \text{ where } i^2 = -1.</math>

In other words, a Gaussian integer is a [[complex number]] such that its [[real part|real]] and [[imaginary part]]s are both [[integer]]s.
Since the Gaussian integers are closed under addition and multiplication, they form a [[commutative ring]], which is a [[subring]] of the field of complex numbers. It is thus an [[integral domain]].

When considered within the [[complex plane]], the Gaussian integers constitute the {{math|2}}-dimensional [[integer lattice]].

The ''conjugate'' of a Gaussian integer {{math|''a'' + ''bi''}} is the Gaussian integer {{math|''a'' − ''bi''}}.

The [[field norm|''norm'']] of a Gaussian integer is its product with its conjugate.
:<math>N(a+bi) = (a+bi)(a-bi) = a^2+b^2.</math>

The norm of a Gaussian integer is thus the square of its [[absolute value]] as a complex number. The norm of a Gaussian integer is a nonnegative integer, which is a sum of two [[square number|squares]]. By the [[sum of two squares theorem]], a norm cannot have a factor <math>p^k</math> in its [[prime decomposition]] where <math>p \equiv 3 \pmod 4</math> and <math>k</math> is odd (in particular, a norm is not itself congruent to 3 modulo 4).

The norm is [[completely multiplicative function|multiplicative]], that is, one has<ref>{{harvtxt|Fraleigh|1976|p=289}}</ref>
:<math>N(zw) = N(z)N(w),</math>
for every pair of Gaussian integers {{math|''z'', ''w''}}. This can be shown directly, or by using the multiplicative property of the modulus of complex numbers.

The [[unit (ring theory)|unit]]s of the ring of Gaussian integers (that is the Gaussian integers whose [[multiplicative inverse]] is also a Gaussian integer) are precisely the Gaussian integers with norm 1, that is, 1, −1, {{math|''i''}} and {{math|−''i''}}.<ref>{{harvtxt|Fraleigh|1976|p=288}}</ref>