# Arithmetics in Extensions of $$\mathbb Q$$

Dušan Đukić

## Abstract

One of the most convenient properties of rational numbers and integers is the uniqueness of factorization into primes. However, the power of the arithmetic in $$\mathbb{Z}$$ is bounded. Thus, some polynomials cannot be factorized into linear polynomials with rational coefficients, but they can always be factorized in a wider field. For instance, the polynomial $$x^2+1$$ is irreducible over $$\mathbb{Z}$$ or $$\mathbb{Q}$$, but over the ring of the so called Gaussian integers $$\mathbb{Z}[i]=\{a+bi\mid a,b\in\mathbb{Z}\}$$ it is factorized as $$(x+i)(x-i)$$. Sometimes the wider field retains many properties of $$\mathbb{Q}$$. We show that some rings, including the Gaussian integers and the ring $$\mathbb{Z}[\omega]$$ ($$\omega$$ being a primitive cubic root of 1), also have the unique factorization property (just like $$\mathbb{Z}$$) and show how they are used. Then we use the latter to prove the Last Fermat Theorem for the exponent n=3.

Arithmetics in the Gaussian Integers $$\mathbb Z[i]$$
Arithmetics in the ring $$\mathbb Z[\omega]$$