Finding factors of factor rings over the Gaussian integers. (English) Zbl 1138.11344

Dresden, Greg; Dymàček, Wayne M.

Am. Math. Mon. 112, No. 7, 602-611 (2005).

From the introduction: The Gaussian integers are defined to be the set \(\mathbb Z[i]= \{a+bi: a,b\in\mathbb Z\), \(i=\sqrt{-1}\}\). These sit inside the complex numbers \(\mathbb C\) and thus obey the usual rules of addition and multiplication; indeed, despite the presence of the imaginary \(i\), they are quite similar to the “traditional” integers. In fact, in the set \(\mathbb Z[i]\) one can define (Gaussian integer) primes, construct analogues of the Euclidean division algorithm and the Euler \(\varphi\)-function, discuss Pythagorean triples, generalize the twin-prime problem, and much more. In this paper, we generalize the idea of factor rings from the integers to the Gaussian integers and discuss what new objects can be found in this manner. (Recall that integer factor rings are the familiar objects \(\mathbb Z/\langle n\rangle\), where \(\langle n\rangle\) signifies the ideal in \(\mathbb Z\) generated by \(n\). These rings are also written as \(\mathbb Z/n\mathbb Z\) or \(\mathbb Z_n\).)

MSC:

11R04	Algebraic numbers; rings of algebraic integers
11R11	Quadratic extensions