Prsten Gaussovih cijelih brojeva i primjene

The ring of Gaussian integers \(\mathbb{Z} [i]\) is a generalization of the ring \(\mathbb{Z}\). As such, Gaussian integers kept most of characteristics of integers. In \(\mathbb{Z} [i]\) we have factorization \( x^2+y^2=(x+yi)(x-yi) \), i=\(\sqrt{-1} \).. Through the properties of invertibility and division, we proved the modified division theorem which helped us prove the division theorem in \(\mathbb{Z} [i]\). Through examples we saw the use of Euclidean algorithm in finding the greatest common divisor of two Gaussian integers. With the help of Bezout’s theorem we showed that \(\alpha, beta \in \mathbb{Z} [i]\) are relatively prime if and only if \(\alpha x + \beta y\) = 1 for some \(x; y \in \mathbb{Z} [i]\). We have talked about primes in \(\mathbb{Z} [i]\). Also, we proved a theorem about recognizing primes in \(\mathbb{Z} [i]\) which tells that if the norm of a Gaussian integer is prime in \(\mathbb{Z}\), then this Gaussian integer is prime in \(\mathbb{Z} [i]\). Moreover, we have shown the unique factorization into primes of the Gaussian integers. At the end, we talked about applications of Gaussian integers. We used Gaussian integers to prove a few claims on primes in Z, describe the (primitive) Pythagorean triples and study the integer solutions of equations \(a^2 + b^2 = c^3\) i \(y^2 + 1 = x^3\).