Editing Gaussian integer (section)

==Primitive residue class group and Euler's totient function==
Many theorems (and their proofs) for moduli of integers can be directly transferred to moduli of Gaussian integers, if one replaces the absolute value of the modulus by the norm. This holds especially for the ''primitive residue class group'' (also called [[multiplicative group of integers modulo n|multiplicative group of integers modulo {{math|''n''}}]]) and [[Euler's totient function]]. The primitive residue class group of a modulus {{math|''z''}} is defined as the subset of its residue classes, which contains all residue classes {{math|{{overline|''a''}}}} that are coprime to {{math|''z''}}, i.e. {{math|(''a'',''z'') {{=}} 1}}. Obviously, this system builds a [[multiplicative group]]. The number of its elements shall be denoted by {{math|''ϕ''(''z'')}} (analogously to Euler's totient function {{math|''φ''(''n'')}} for integers {{math|''n''}}).

For Gaussian primes it immediately follows that {{math|''ϕ''(''p'') {{=}} {{abs|''p''}}<sup>2</sup> − 1}} and for arbitrary composite Gaussian integers
:<math>z = i^k\prod_m {p_m}^{\nu_m}</math>
[[Euler's totient function|Euler's product formula]] can be derived as
:<math>\phi(z) =\prod_{m\, (\nu_m > 0)} \bigl|{p_m}^{\nu_m}\bigr|^2 \left( 1 - \frac 1{|p_m|{}^2} \right) = |z|^2\prod_{p_m|z}\left( 1 - \frac 1{|p_m|{}^2} \right)</math>
where the product is to build over all prime divisors {{math|''p<sub>m</sub>''}} of {{math|''z''}} (with {{math|''ν<sub>m</sub>'' > 0}}). Also the important [[Euler's theorem|theorem of Euler]] can be directly transferred:
: For all {{math|''a''}} with {{math|(''a'',''z'') {{=}} 1}}, it holds that {{math|''a''<sup>''ϕ''(''z'')</sup> ≡ 1 (mod ''z'')}}.