Olympiad training materials (main page)

# 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