Every mobile phone, CD player, and computer contains a chip that solves linear equation systems over a finite field at lightning speed to correct erroneous data. This book explains the mathematical inner workings of such a chip. Finite fields are number systems (so-called Galois fields) with only a finite number of elements, which can be added, subtracted, multiplied, and divided. The main aim of the book is to explain and practice how these calculations are performed in an elementary way. It is aimed at anyone who is familiar with mathematical language and wants to understand how finite fields operate. A certain familiarity with basic concepts of linear algebra, as practiced in an engineering mathematics course, is assumed. Although the text is focused, it also provides a basic introduction.