Olympiad training materials (main page)

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.

No. | Title and link |
---|---|

1. | General properties |

2. | Arithmetics in Gaussian integers |

3. | Arithmetics in \(\mathbb Z[\omega]\) |

4. | Arithmetics in other quadratic rings |