ABSTRACT
After the work of the ancient Greek and later geometers, it is customary to parametrize the Euclidean line by the algebra ℝ of real numbers. The real numbers ℝ contain the rational numbers ℚ, and in particular, the (rational) integers ℤ that are the subject of number theory. In particular, they satisfy the unique factorization theorem whose traditional statement is that each positive integer is a product of (positive) prime numbers in a way that is unique up to order. An ideal of ℤ is a subset I with the following properties: 0∈I; the sum of any two elements in I is in I; and all of the multiples of any member of I by arbitrary integers are again in I. It is a principal ideal only if it consists of all multiples of some integer g, called the generator. The chapter also provides an overview of the key concepts discussed in this book.
