Introduction to Commutative Rings

Commutative rings are one of the most fundamental structures in algebra. In particular, the integers, the rational numbers, the real field and the complex field are all commutative rings. Many other objects such as polynomial rings, matrix rings, vector spaces, and modules are constructed from these.

Here only the principal fields supported will be noted. The field of rational numbers ℚ is the starting point for constructing a large number of commutative rings, the most important of which is the ring of integers ℤ. Another basic field is a finite field F_q, where q = p^k for some prime p.

Global fields comprise two families of fields:

• A number field is a finite extension K of ℚ. The ring of integers O_K of a number field K is a Dedekind domain and is the main focus in the study of number fields.
• A global function field is a finite extension of the field F_q(t); that is, the field of rational functions over the finite field F_q.

A great deal of machinery has been constructed for working with global fields. This has been developed jointly by the KANT group (TU, Berlin) and the Magma group. Magma is the only widely used system that provides machinery for global function fields, which are of great importance in the study of the arithmetic properties of curves. In fact the machinery for function fields also allows function fields that are extensions of the rational function field ℚ(t).

The second important class of fields are the local fields which are completions of the global fields:

• The field of real numbers ℝ and the field of complex numbers ℂ.
• The field of p-adic numbers ℚ_p, p a prime, and their finite extensions.
• The field of formal Laurent series F_q((t)) over a finite field F_q.

Many calculations are assumed to take place in the algebraic closure of a field. Mostly, it is possible to choose a suitable finite extension at the outset and so there is no need to define the algebraic closure. However, for cases where this is not possible, the formal algebraic closures of ℚ and F_q can be defined. Such fields are represented by polynomials defining the generating elements of the current approximation to the closure and most operations are performed using Gröbner basis techniques.

In this chapter only the integers, finite fields, the real and complex, fields and polynomial rings are considered. Information about global and local fields can be found under Number Theory. In the final section an attempt is made to summarise the commutative rings supported in Magma by classifying them according to the axioms they satisfy.