讲座内容：

In 1847, French mathematician Gabriel Lamé claimed to have proved Fermat's Last Theorem. However, he mistakenly assumed that the ring Z[ζn], where ζn is an n-th root of unity, is a unique factorization domain for all n. This error led to the discovery of the class group of a Dedekind domain, and ultimately to the development of factorization theory.

We will provide an overview of factorization theory in integral domains and monoids. After discussing about arithmetical invariants, we will give the definition of a Krull monoid and explain how the class group of a Krull monoid H can be used to describe the arithmetic of H through combinatorial methods.

Our main focus will be on certain algebraic number rings, known as orders in number fields. These objects exhibit more complicated arithmetic behavior than Krull monoids and we will show how, in certain cases, one can describe the arithemtic of an order using its algebraic properties.