### **2.13. Number Fields

In this section, we develop the theory of number fields and rings. Our aim is to make accessible to the readers the working of the cryptanalytic algorithms based on number field sieves.

#### 2.13.1. Some Commutative Algebra

Commutative algebra is the study of commutative rings with identity (rings by our definition). Modern number theory and geometry are based on results from this area of mathematics. Here we give a brief sketch of some commutative algebra tools that we need for developing the theory of number fields.

##### Ideal arithmetic

We start with some basic operations on ideals (cf. Example 2.7, Definition 2.23).

##### Definition 2.92.

Let A be a ring and let , , be a family (not necessarily finite) of ideals in A.
The set-theoretic ... |