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.

Table of Contents

General properties

Arithmetics in the Gaussian Integers \( \mathbb Z[i] \)

Arithmetics in the ring \( \mathbb Z[\omega] \)

Arithmetics in other quadratic rings


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