A Gaussian integer is a complex number whose real and imaginary part are both integers . That is aGaussian integer is a complex number of the form a +ib where a and b are integers.The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i].
Formally, Gaussian integers are the set
\mathbb{Z}[i]=\{a+bi \mid a,b\in \mathbb{Z} \}.
Thse absolute value of Z= a+ib is √a2 + b2 .The square of the absolute value is called the numbers complex norm.
Norm (Z)=a2 + b2
For example, N(2+7i) = 22 +72 = 53.
The norm is multiplicative i.e.
N(z\cdot w) = N(z)\cdot N(w).
The only Gaussian integers which are invertible in Z[i] are 1 and i.
The units of Z[i] are therefore precisely those elements with norm 1, i.e. the elements
1, −1, i and −i.
Divisibility in Z[i] is de ned in the natural way: we say β divides α if
α = βγ for some
γ ε Z[i]. In this case, we call a divisor or a factor of .
A Gaussian integer = a + bi is divisible by an ordinary integer c if and
only if c divides a and c divides b in Z.
A Gaussian integer has even norm if and only if it is a multiple of 1 + i.
Historical background
The ring of Gaussian integers was introduced by Carl Friedrich Gauss in his second monograph on (1832). The theorem of quadratic reciprocity (which he had first succeeded in proving in 1796) relates the solvability of the congruence x2 ≡ q (mod p) to that of x2 ≡ p (mod q). Similarly, cubic reciprocity relates the solvability of x3 ≡ q (mod p) to that of x3 ≡ p (mod q), and biquadratic (or quartic) reciprocity is a relation between x4 ≡ q (mod p) and x4 ≡ p (mod q). Gauss discovered that the law of biquadratic reciprocity and its supplements were more easily stated and proved as statements about "whole complex numbers" (i.e. the Gaussian integers) than they are as statements about ordinary whole numbers (i.e. the integers).
