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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.

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